Trajectory planning of a closed five-bow-shaped bar linkage based on finite Fourier series

Abstract

With aim to reduce the energy consumption, a trajectory planning method is presented for a closed five-bow-shaped bar linkage, which can be propelled itself by morphing configuration. The objective herein is to optimize the driving joints trajectories within the global feasible region when the linkage rolls along the ground with a desired acceleration. The driving joint trajectories were represented by finite Fourier series, whose coefficients were solved by genetic algorithm to ensure a minimal energy consumption of the linkage. The impact of the number of terms of finite Fourier series on the energy consumption was also discussed through numerical examples. As a result, the energy consumption based on this strategy had been reduced by 19%, comparing with the constant potential energy strategy. A number of terms between six and eight using to denote the joint trajectories are appropriate, because that a small number of terms is incapable of expressing the joint trajectories accurately, whereas, a large number makes the joints to be subjected to vibration shock. At last, simulation on a virtual model and experiments on a prototype were carried out to verify the effectiveness of the proposed method.

Introduction

The application of mobile robots, which usually use mobile mechanisms such as wheel, track, foot, and wheel–foot composite methods, is becoming increasingly widespread. ¹ Bionics research has shown that there are many motion modes in nature that use limbs as rolling bodies.^2,3 Spherical robots, planar rolling linkages, and spatial polyhedral rolling robots are becoming a research focus.^4–6 In our research project, we have investigated a closed five-bow-shaped bar linkage which can roll along a straight line. It can achieve actions such as accelerated rolling, climbing, and obstacle crossing by changing its shape to generate gravity bias torque.^7–9 In order to reduce the energy consumption of the closed five-bow-shaped bar linkage during the rolling process, we have established the kinematics and dynamics models of the linkage and proposed an energy–optimal trajectory planning method.

Trajectory planning is a core technology in mobile robot control, as the quality of trajectory has a significant impact on the motion performance of robots. In different application scenarios, the objectives of trajectory planning, no matter in operation space or joint space, can be roughly divided into time optimization, energy optimization, kinematics performance optimization, and comprehensive performance optimization.^10–12 Trajectory planning in operation space is prone to the problems, such as kinematic singularity, discontinuous acceleration, and excessive jumping. It is computationally expensive, because that it needs to determine the path of end-effector and followed by inverting the path into joint space.^13,14 Conversely, trajectory planning in joint space plan the joint trajectories directly, that can avoid kinematic singular problems effectively with low computational cost and also ensure the continuity of joint motion. ¹⁵ Therefore, trajectory planning in joint space has drawn much attention due to its advantages. Rubio et al. ¹⁶ proposed a method to plan joint trajectory based on interpolation functions, that first calculates a joint’s trajectory with minimal time consumption and then uses iterative algorithms to ensure the trajectory meet obstacle crossing requirements. Bureerat et al. ¹⁷ studied the multi-objective trajectory-planning problem of a 6-DOF robot, in which the joint trajectory with minimal time and impact was calculated out with the constraints of displacement, velocity, and acceleration. Ekrem and Aksoy ¹⁸ proposed a time-optimal joint space trajectory planning method for a manipulator to meet with the needs of multi-path points planning. In their study, the trajectory between the starting and ending points was carried out using particle swarm optimization (PSO) algorithm. This method effectively improved trajectory tracking accuracy and reduced the vibration of manipulator by using the fifth-order polynomial.

In fact, trajectory planning of robots can be regarded as optimization problems, and various intelligent algorithms have been employed to obtain the optimal solution.^19,20 In the study of Kucuk for Fully Planar Parallel Manipulators, a scheduling algorithm was added to Proposed Numerical Algorithm (PNA) in order to find the best trajectory in terms of dexterity between two points, while PSO algorithm was also used for cubic spline optimization with the aim to have time optimal smooth motion. ²¹ Huang et al. ²² used the elitist non-dominated sorting genetic algorithm to address the time-jerk optimal trajectory planning problem, in which the influences of weights of two subgoals, that is, time and impact, were discussed and then the trajectory was interpolated in the joint space by means of fifth-order B-spline after the Pareto optimal front and optimal solution were acquired. In the work of Zhang et al., ²³ Cubic splines were used to interpolate in joint space, and the optimal time-jerk trajectory of a hydraulic robotic excavator was obtained using the sequential quadratic programming algorithm with the constraints of joint angular velocity, angular acceleration, and also jerk. Ni et al. ²⁴ used PSO algorithm to plan the obstacle avoidance trajectory of a dual-arm space robot. The joint trajectories thereof were denoted by a sine function with seventh-order polynomial parameters, and several penalty items were used so that the robot could complete specified goals. In the study of Wang et al. for a six-axis grinding robot, improved whale optimization algorithm and differential evolution algorithm were used to meet the requirements of minimum time and minimum jerk. ²⁵ After optimization, the jerk of every joint has reduced significantly, which can improve the grinding quality to some extent. In the study of Ege and Kucuk, PSO algorithm combined with a discrete-time PID controller was applied to reduce energy consumption of above-knee prosthesis, resulting in a 51% reduction in energy consumption. ²⁶

The trajectory planning methods mentioned above may obtain the optimal parameters for a certain type of trajectory, rather than all types. The parameters acquired by these methods are relative optimum, because that they cannot denote universal motion law of joint trajectory. Then, Fourier series was introduced to plan joint trajectory, as it can represent any type of trajectory using infinite series of sine and cosine functions. Yang et al. ²⁷ studied the foot trajectories based on Fourier series to reduce the joints energy consumption of a quadruped robot. And only the first two cosine and sine terms as well as the constant term were considered in their work, in order to reduce calculation time. Caruso et al. ²⁸ presented a procedure to generate an approximate time-optimal trajectory of a solar sail through a finite Fourier series (FFS). Due to its simplicity and effectiveness, the proposed procedure can be used to obtain an optimal trajectory rapidly when deal with fast tasks. In the work of Chen et al., ²⁹ the sheave rotation motion of a mechatronic elevator system was described using Fourier sine series (FSS) whose coefficients were optimized through a real-coded genetic algorithm with aim to minimize input energy.

Based on the kinematics and dynamics models of the closed five-bow-shaped bar linkage established before, Lagrange’s equation with a dissipation function was used to obtain the energy consumption in this paper. And the following studies were committed to the joint space trajectory planning method: (1) A joint trajectory representing method that using FFS to denote any joint path of the proposed linkage in the global feasible region. (2) Optimization function, which has the objective of minimum energy consumption and also several motion constraints, was established to obtain the coefficients of driving joint trajectories through Genetic algorithm. (3) The impact of the number of terms intercepted from Fourier series to denote the joint trajectories on the energy consumption was discussed through numerical examples. (4) Simulation on a virtual model and experiments on a physical prototype were used to verify the effectiveness of proposed trajectory planning method.

Dynamic model of the closed five-bow-shaped bar linkage

Description of the mechanism

According to the mechanical structure, robots can be divided into three categories: serial robots, parallel robots, and hybrid robots. The serial robot consists of a set of open-loop kinematic chains, the parallel robot consists of some closed-loop kinematic chains, while the hybrid robot includes both of them. Parallel robots have the advantages of fast speed, small mass, and high accuracy, but solving their inverse Jacobian matrices is very complex. In the study of Kucuk, general equations for the inverse Jacobian matrices of 195 6-DOF GSP mechanisms were symbolically derived, and two different mechanisms (symmetrical and asymmetrical) were also given as examples to describe the methodology. ³⁰ The closed five-bow-shaped bar linkage was derived from the conventional planar five-bar mechanism, which has five straight bars while one of them is used as frame fixed link. Then, the straight bars were replaced with bow-shaped bars with circular outer contours, and the integrated coupling of the frame fixed link was freed. In this way, a closed five-bow-shaped bar linkage that can roll straight is obtained. Although the fixed link has been freed, the closed five-bow-shaped bar linkage still has some advantages of parallel robots, including high accuracy, compact structure, symmetry, and isotropy. It also solves the problems of low degrees of freedom and rigid motion modes in traditional parallel mechanisms, and can be used as a multi degree of freedom rolling mechanism. As shown in Figure 1, the linkage was modular designed and the outer contour of each module is one-fifth of a full circle. Each bow module is mainly composed of left and right bow-shaped plates, a spindle, a motor, and bevel gears. In order to ensure the structural symmetry and weight balance, a DC motor is installed in each bow module. The outer sides of the left and right bow plates are connected to the spindle through a flange, while the inner sides are connected to the spindle through bearings. So that, each bow module can rotate around the joint on either side. The supporting parts are mainly used for fixing the motor, preventing vibration during operation, and achieving the connection of left and right bow-shaped plates. The output shaft of the motor is perpendicular to the joint shaft, where bevel gear transmission is used to change the direction of rotation and maintain a compact structure.

Figure 1.

The structure of the closed five-bow-shaped bar linkage: (a) overall structure and (b) driving joint. 1 – Left bow-shaped plate; 2 – Right bow-shaped plate; 3 – Flange; 4 – Spindle; 5 – Bevel gear; 6 – Supporting part; 7 – Motor.

By installing a mass block in each bow module, we ensured that each module has the same mass with the barycenter located at the midpoint of the line connecting the adjacent joints. Through the gravity bias torque caused by the regular motion of driving joints, the closed five-bow-shaped bar linkage can complete continuous rolling motion along a straight line without rigid impact, due to the smooth contact with the ground.

Kinematics model of linear rolling

According to the structure of the closed five-bow-shaped bar linkage, the kinematics model of linear motion for the linkage was established, as shown in Figure 2.

Figure 2.

Kinematics model.

As shown as Figure 2, we assume that the joints of the closed five-bow-shaped bar linkage are marked as A, B, C, D, and E, and the joint A is in contact with the ground point $O$ at the initial rolling moment. Then, the point $O$ is taken as the origin to establish the ground coordinate system $\mathrm{OXY}$ , which has the $X$ -axis straight along the rolling direction and the $Y$ -axis upward perpendicular to the ground. The local coordinate system $\mathrm{oxy}$ is established with the origin in accord with the curvature center of the outer contour of touchdown bar, and the directions of its axes are always as same as $\mathrm{OXY}$ . By extending the outer contour of touchdown bar, a virtual circle whose center coincides with the origin of $\mathrm{oxy}$ can be drawn out. Link coordinate systems $x_i{y}_i(i=1,2,\dots ,5)$ are established at the joints A, B, C, D, and E, whose $x_i$ -axes are straight along the chords of the bow shaped bars with the directions toward from the former joint to the latter. P is the current touchdown point, then, the angle between $\mathrm{oA}$ and $\mathrm{oP}$ can be defined as the roll angle $\varphi$ of the linkage. The angles between $x_i$ and $x_{i-1}$ are defined as joint angle ${\theta }_i(i=1,2,\dots ,5)$ . R and $m_i(i=1,2,\dots ,5)$ represent the curvature radius and mass of the bow-shaped bar, respectively. Due to modular design and balance weight, each bow-shaped bar has the same mass and the centroid locates at the midpoint of the chord of the bow-shaped bar.

During the linear rolling process of the closed five-bow-shaped bar linkage, the five bowed bars touch the ground in sequence. Then, the rolling process can be divided into five stages according to the grounded bar. Therefore, we can only analyze the joint trajectories planning method for any one stage, due to the symmetrical design. For instance, the rolling stage when $\varphi \in [0,2\mathrm{\pi }/5]$ is studied in this paper, and the planning methods for other stages are as same as this case.

When bar 1 touches the ground, the angle between $x_1$ and the ground is denoted as ${\psi }_1$ .

$$\begin{array}{c}{\psi }_1=\frac{\pi }{5}-\varphi \hfill \end{array}$$

(1)

Then, the angle between $x_i(i=2,3,\dots ,5)$ and the ground can be expressed as ${\psi }_i$ .

$$\begin{array}{c}{\psi }_i={\psi }_1+\sum _{i=2}^i{\theta }_i,i=2,3,\dots ,5\hfill \end{array}$$

(2)

The coordinates of the centroid of bar $i$ with respect to $\mathrm{OXY}$ can be respectively denoted as $X_i$ and $Y_i$ , and

$$\{\begin{array}{c}{X}_i=X_A+\sum _{j=1}^{i-1}\left[lcos{\psi }_j+\frac{l}{2}cos{\psi }_i\right]\\ Y_i=Y_A+\sum _{j=2}^i\left[lsin{\psi }_j+\frac{l}{2}sin{\psi }_i\right]\end{array}$$

(3)

where, $l$ represents the chord length of bow-shaped bar and $l=2R\sin (\pi /5)$ , $X_A$ and $Y_A$ represent the coordinates of joint $\mathrm{A}$ with respect to $\mathrm{OXY}$ , and

$$\begin{array}{c}\{\begin{array}{c}{X}_A=R\varphi -\mathrm{Rsin}(\frac{\pi }{5}-{\psi }_1)\hfill \\ Y_A=R-\mathrm{Rcos}(\frac{\pi }{5}-{\psi }_1)\hfill \end{array}\hfill \end{array}$$

(4)

As the translation transformation between the coordinate systems $\mathrm{OXY}$ and $\mathrm{oxy}$ , rather than rotation transformation, the coordinates of the centroid of the whole linkage relative to the local coordinate system $\mathrm{oxy}$ could be calculated by

$$\begin{array}{c}\{\begin{array}{c}{x}_s=\frac{\sum _{i=1}^5{m}_i(X_i-R\varphi )}{m_t}=\frac{1}{5}\sum _{i=1}^5(X_i-R\varphi )\hfill \\ y_s=\frac{\sum _{i=1}^5{m}_i(Y_i-R)}{m_t}=\frac{1}{5}\sum _{i=1}^5(Y_i-R)\hfill \end{array}\hfill \end{array}$$

(5)

where, $x_s$ and $y_s$ represent the coordinates of the centroid of the whole linkage relative to $\mathrm{oxy}$ , $m_t$ represents the mass of the whole linkage and $m_t=5m_i$ . Taking the rolling stage when $\varphi \in [0,2\mathrm{\pi }/5]$ as an example, the coordinates of the whole linkage’s centroid could be obtained as below.

$$\begin{array}{c}\{\begin{array}{c}{x}_s=-\frac{3}{5}{e}_1{c}_{\alpha }-e_2{s}_{\alpha }-\frac{6}{5}{e}_1{c}_{\alpha +{\theta }_2}-\frac{4}{5}{e}_1{c}_{\alpha +{\theta }_{23}}-\frac{2}{5}{e}_1{c}_{\alpha +{\theta }_{234}}\hfill \\ y_s=-\frac{3}{5}{e}_1{s}_{\alpha }+e_2{c}_{\alpha }-\frac{6}{5}{e}_1{s}_{\alpha +{\theta }_2}-\frac{4}{5}{e}_1{s}_{\alpha +{\theta }_{23}}-\frac{2}{5}{e}_1{s}_{\alpha +{\theta }_{234}}\hfill \end{array}\hfill \end{array}$$

(6)

where, $\alpha =\pi /5-\varphi$ , $e_1=-R\sin (\mathrm{\pi }/5)$ , $e_2=-R\cos (\mathrm{\pi }/5)$ , $s_i=\sin {\theta }_i$ , $c_i=\cos {\theta }_i$ , ${\theta }_{\mathrm{ij}}={\theta }_i+{\theta }_j$ , and so on.

We assume that the linear rolling of the closed five-bow-shaped bar linkage is pure rolling with touchdown point $P$ as the instantaneous center of velocity. Then, the rolling movement of the linkage could be regarded as a rotation motion around the instantaneous axis which is pass through point $P$ and perpendicular to motion plane. And the rotational inertia of the whole linkage around point $P$ can be calculated by

$$\begin{array}{c}{J}_P=\sum _{i=1}^5[J_i+m_i{(X_i-R\varphi )}^2+m_i{Y_i}^2]\hfill \end{array}$$

(7)

where, $J_i(i=1,2,\dots ,5)$ represent the rotational inertias of the bow shaped bars around their respective center of mass.

The rolling acceleration of the linkage is mainly attributed to gravity bias torque by ignoring the influence of inertia torque. Then the equilibrium equation could be illustrated as

$$\begin{array}{c}{J}_P({\theta }_1,{\theta }_2,\varphi )\stackrel{··}{\varphi }=m_{t}g{x}_s({\theta }_1,{\theta }_2,\varphi )\hfill \end{array}$$

(8)

where, $g$ represents the gravitational acceleration.

Dynamic equation considering joint friction

In fact, mechanical systems are inevitably affected by the characteristics of the joints, especially for the joint viscous damping which may lead to joint friction torque, hinder joint movement, result in the increasement of energy consumption. And the dissipation function of the closed five-bow-shaped bar linkage caused by viscous friction could be illustrated as

$$\mathrm{\Psi }\left(\mathit{q},\stackrel{.}{\mathit{q}}\right)=\frac{1}{2}\sum _{i=1}^5{f}_i{\stackrel{.}{\theta }}_i^2$$

(9)

where, $f_i(i=1,2,\dots ,5)$ represent the coefficients of viscous friction in the joints, $\mathit{q}={[{\mathit{\theta }}_1,{\mathit{\theta }}_2,\varphi ]}^{\mathit{T}}$ represents the generalized coordinate vector using to describe the linkage, $\stackrel{·}{\mathit{q}}=[{\stackrel{·}{\theta }}_1,{\stackrel{·}{\theta }}_2,\stackrel{·}{\varphi }{]}^T$ represents the generalized velocity vector.

As a typical multi-rigid-body system, the total kinetic energy of the closed five-bow-shaped bar linkage is composed of the kinetic energy of each bar. Therefore, the total kinetic energy $T$ can be calculated by

$$T=\frac{1}{2}\sum _{i=1}^5\left(m_i{{\stackrel{.}{X}}_i}^2+m_i{{\stackrel{.}{Y}}_i}^2+J_i{{\stackrel{.}{\psi }}_i}^2\right)$$

(10)

where, ${\stackrel{·}{X}}_i$ and ${\stackrel{·}{Y}}_i$ represent velocity components of the centroid of bar $i$ with respect to $\mathrm{OXY}$ , ${\stackrel{·}{\psi }}_i$ represents the rotational angular velocity of bar $i$ around touchdown point P and

$$\begin{array}{c}{\stackrel{·}{\psi }}_i=\{\begin{array}{c}\stackrel{·}{\alpha },i=1\hfill \\ \stackrel{·}{\alpha }+{\sum }_{j=2}^i{\stackrel{·}{\theta }}_j,2\le i\le 5\hfill \end{array}\hfill \end{array}$$

(11)

To define the ground as the zero-potential-energy reference surface, the total potential energy of the linkage can be denoted as

$$\begin{array}{c}V=m_{t}g(y_s+R)\hfill \end{array}$$

(12)

Then, the Lagrange’s equation of the closed five-bow-shaped bar linkage can be established as

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \stackrel{.}{\mathit{q}}}\right)-\frac{\partial L}{\partial \mathit{q}}+\frac{\partial \mathrm{\Psi }}{\partial \stackrel{.}{\mathit{q}}}=\tau$$

(13)

where, $\mathit{\tau }=[{\tau }_1,{\tau }_2,{\tau }_{\varphi }{]}^T$ represents the generalized torque vector corresponding to the generalized coordinate vector, $L$ represents Lagrange function and $L=T-V$ .

This Lagrange’s equation also can be written in matrix form, as shown as below

$$M\left(\mathit{q}\right)\stackrel{··}{\mathit{q}}+H\left(\mathit{q},\stackrel{.}{\mathit{q}}\right)+G\left(\mathit{q}\right)+D\left(\mathit{q}\right)\stackrel{.}{\mathit{q}}=\tau$$

(14)

where, $\mathit{M}(\mathit{q})$ represents the inertia matrix; $\mathit{H}(\mathit{q},\stackrel{·}{\mathit{q}})$ represents the torque term generated by the centrifugal force; $\mathit{G}(\mathit{q})$ represents the torque term generated by the influence of gravity, $\mathit{D}(\mathit{q})$ represents the damping coefficient matrix. Because that the linkage is subjected to a conservative force on the generalized coordinate $\varphi$ , the generalized torque vector is $\mathbf{\tau }\mathbf{=}\mathbf{[}{\mathbf{\tau }}_1\mathbf{,}{\mathbf{\tau }}_2\mathbf{,}0{\mathbf{]}}^{\mathbf{T}}$ .

Motion representation based on Fourier series

There are many methods used to plan the trajectory in joint space, such as polynomial motion laws, B-spline curves, trigonometric splines, etc. Among them, Polynomial curves require a large amount of computation; the derivative of a B-spline curve exhibits a sudden change at the starting and ending points; trigonometric functions may cause joint angle singularity. The trajectories generated by these methods could be close to the real paths in a specified task. However, they cannot infinitely close to the true value, due to the characteristics of the motion functions used to denote the joint motions in these methods. Then, the optimization of trajectory parameters is operated within a local scope, rather than global area. In other words, it is possible that the trajectory obtained by the aforementioned methods is still not the best one to accord with the optimization purpose.

However, Fourier series can represent any continuous periodic function as a linear combination of trigonometric functions. And it has been proved to be a useful tool of trajectory planning, because of its simple form and easy processing feature. Using Fourier series to characterize the angular displacements of driving joints can help to establish the precise mathematical expressions of joint trajectories. The more important is that it makes the optimization of trajectory operate in the whole feasible region with the target of lowest energy consumption.

When the closed five-bow-shaped bar linkage moves according to a given acceleration, its motion can be described by the displacements of the driving joints over time. We assumed that the angles of driving joints change from ${\theta }_i(0)$ to ${\theta }_i(T_0)$ while the time interval is $t=[0,T_0]$ . Then, the joint angles of the linkage during the rolling process are subject to the following constraints:

$$\{\begin{array}{c}{\theta }_i\left(0\right)={\theta }_0,{\theta }_i\left(T_0\right)={\theta }_f\\ {\stackrel{.}{\theta }}_i\left(0\right)={\stackrel{.}{\theta }}_0,{\stackrel{.}{\theta }}_i\left(T_0\right)={\stackrel{.}{\theta }}_f\\ {\stackrel{··}{\theta }}_i\left(0\right)={\stackrel{··}{\theta }}_0,{\stackrel{··}{\theta }}_i\left(T_0\right)={\stackrel{··}{\theta }}_f\end{array}$$

(15)

Fourier series is used to describe the joint trajectories in the motion period of $t=[0,T_0]$ . And then, ${\theta }_i(t)$ can be expressed as

$${\theta }_i\left(t\right)=a_0+\sum _{n=1}^{\infty }\left[a_{n}cos\left(n{\omega }_{0}t\right)+b_{n}sin\left(n{\omega }_{0}t\right)\right]$$

(16)

where, $a_n$ and $b_n$ represent the coefficients of the Fourier series; $a_0$ represents the constant term; $n$ represents the number of terms, ${\omega }_0$ represents the first harmonic of the Fourier series, and ${\omega }_0=2\mathrm{\pi }/T_0$ .

Further, the angular velocities of joints, as well as angular accelerations, can be obtained by derivation. Then, the angular velocities and accelerations can be illustrated as

$${\stackrel{.}{\theta }}_i\left(t\right)=\sum _{n=1}^{\infty }\left[n{\omega }_0{a}_{n}cos\left(n{\omega }_{0}t+\pi /2\right)+n{\omega }_0{b}_{n}sin\left(n{\omega }_{0}t+\pi /2\right)\right]$$

(17)

$${\stackrel{··}{\theta }}_i\left(t\right)=\sum _{n=1}^{\infty }\left[n^2{{\omega }_0}^2{a}_{n}cos\left(n{\omega }_{0}t+\pi \right)+n^2{{\omega }_0}^2{b}_{n}sin\left(n{\omega }_{0}t+\pi \right)\right]$$

(18)

Fourier series has the ability to fit any type of joint trajectory due to its infinite series. However, the coefficients of Fourier series are hardly obtained completely. It is necessary to intercept a finite number of terms of Fourier series, with aim to acquire approximate optimal result with a low computational cost. Assuming that the number of intercepted terms is $M$ , the infinite Fourier series may be instead by finite Fourier series (FFS) with the terms of $M$ . Then, the displacements, angular velocities, and angular accelerations of the joints can be written in the form of FFS, and

$$\begin{array}{c}{\theta }_i\left(t\right)=a_0+\sum _{n=1}^M[a_n\cos \left(n{\omega }_{0}t\right)+b_n\sin \left(n{\omega }_{0}t\right)]\hfill \end{array}$$

(19)

$${\stackrel{.}{\theta }}_i\left(t\right)=\sum _{n=1}^M\left[n{\omega }_0{a}_{n}cos\left(n{\omega }_{0}t+\pi /2\right)+n{\omega }_0{b}_{n}sin\left(n{\omega }_{0}t+\pi /2\right)\right]$$

(20)

$${\stackrel{··}{\theta }}_i\left(t\right)=\sum _{n=1}^M\left[n^2{{\omega }_0}^2{a}_{n}cos\left(n{\omega }_{0}t+\pi \right)+n^2{{\omega }_0}^2{b}_{n}sin\left(n{\omega }_{0}t+\pi \right)\right]$$

(21)

Trajectory planning

Planning objective

During the robot trajectory planning, the time, energy, dynamics, and also comprehensive performance are usually taking into consideration. As the closed five-bow-shaped bar linkage rolls along the ground in a given acceleration, it is suitable to take energy consumption as the planning objective in order to acquire the optimal joint trajectory with minimum consumption during the linear rolling.

Joints A and B are selected as driving joints in the rolling stage when $\varphi \in [0,2\mathrm{\pi }/5]$ . Then, the total energy consumption $Q$ , including joint friction, can be expressed as

$$Q=\underset{0}{\overset{T_0}{\int }}\left(P_1+P_2\right)dt=\underset{0}{\overset{T_0}{\int }}\left({\tau }_1{\stackrel{.}{\theta }}_1+{\tau }_2{\stackrel{.}{\theta }}_2\right)dt$$

(22)

where $P_1$ and $P_2$ represent the instantaneous power of joint A and joint B, respectively.

When calculating the definite integral, the Newton Leibniz formula is usually applicable. However, for equation (22), it is difficult to express the driving joint power as a function during the movement process, and numerical methods need to be used to obtain approximate results. And then, we use the rectangular method to divide the driving joint power within ${\mathrm{T}}_0$ seconds into 200 intervals, so the function value can be calculated using the left point of each interval. By calculating the area of all rectangles and summing them up, the total energy consumption of the system during motion can be obtained.

Constraints

The five bow-shaped bars contact the ground in sequence during the rolling process. The switching of grounded bar will lead the linkage to be suffered by the impact force from the ground. Therefore, Wang et al. ⁷ established three constraints to avoid the impact force from the ground and ensure stability and continuity of the rolling motion. The three constraints can be described as follows.

• (1) The inner contour of the closed linkage must be kept as a convex pentagon to avoid kinematic singular problems. Thus, the angles of the driving joints should be maintained within the range of $[0°,151°]$ , according to the polygon external angle sum theorem.

• (2) The angle constraint for the joint which is just contacting with ground is strict, because that the linkage is confronted with the alternation of touchdown bars at the moment. When a joint comes into contact with ground, the outer contour curves of the two bars near this grounded joint should be continuous. Therefore, the joint angle of the touchdown joint should be equal to 72° at the critical position, in order to ensure a smooth transition between different motion stages.

• (3) There should be only one touchdown point between the linkage and the ground. Otherwise, the stability and continuity of the rolling motion will be disrupted.

According to the three constraints, the motion range of driving joint angles for any roll angle can be calculated out.

As shown as Figure 3(a), when $\varphi \in [0,\pi /5)$ and ${\theta }_1\in (2\pi /5-2\varphi ,\mathrm{\pi }-\arccos (1/4)]$ , ${\theta }_2$ should meet the following requirements:

$$\begin{array}{c}\{\begin{array}{c}{\theta }_{2\min }=\pi -\frac{{\theta }_1}{2}-\arccos \left(\frac{2\cos {\theta }_1-1}{2\sqrt{2+2\cos {\theta }_1}}\right)\hfill \\ {\theta }_{2\max }=\pi -\frac{{\theta }_1}{2}-\arccos \left(\frac{2\cos {\theta }_1+5}{4\sqrt{2+2\cos {\theta }_1}}\right)\hfill \end{array}\hfill \end{array}$$

(23)

Figure 3.

Joint angle constraints when $\varphi \in [0,\mathrm{\pi }/5)$ : (a) ${\theta }_1\in (2\pi /5-2\varphi ,\mathrm{\pi }-\arccos (1/4)]$ and (b) ${\theta }_1\in (\mathrm{\pi }-\arccos 1/4,\mathrm{\pi }-2\arcsin 1/4)$ .

As shown as Figure 3(b), when $\varphi \in [0,\pi /5)$ and ${\theta }_1\in (\mathrm{\pi }-\arccos 1/4,\mathrm{\pi }-2\arcsin 1/4)$ , ${\theta }_2$ should meet the following requirements:

$$\begin{array}{c}\{\begin{array}{c}{\theta }_{2\min }=0\hfill \\ {\theta }_{2\max }=\pi -\arccos \left(\frac{2\cos {\theta }_1+1}{\sqrt{5+4\cos {\theta }_1}}\right)-\arccos \left(\frac{\sqrt{5+4\cos {\theta }_1}}{2}\right)\hfill \end{array}\hfill \end{array}$$

(24)

As shown as Figure 4(a), when $\varphi \in [\mathrm{\pi }/5,2\mathrm{\pi }/5]$ and ${\theta }_2\in (2\pi /5-2\varphi ,\mathrm{\pi }-\arccos (1/4)]$ , ${\theta }_1$ should meet the following requirements:

$$\begin{array}{c}\{\begin{array}{c}{\theta }_{1\min }=0\hfill \\ {\theta }_{1\max }=\pi -\arccos \left(\frac{2\cos {\theta }_2+1}{\sqrt{5+4\cos {\theta }_2}}\right)-\arccos \left(\frac{\sqrt{5+4\cos {\theta }_2}}{2}\right)\hfill \end{array}\hfill \end{array}$$

(25)

Figure 4.

Joint angle constraints when $\varphi \in [\pi /5,2\pi /5]$ : (a) ${\theta }_2\in (2\pi /5-2\varphi ,\mathrm{\pi }-\arccos (1/4)]$ and (b) ${\theta }_2\in (\mathrm{\pi }-\arccos (1/4),\mathrm{\pi }-2\arcsin (1/4)]$ .

As shown as Figure 4(b), when $\varphi \in [\mathrm{\pi }/5,2\mathrm{\pi }/5)$ and ${\theta }_2\in (\mathrm{\pi }-\arccos (1/4),\mathrm{\pi }-2\arcsin (1/4)]$ , ${\theta }_1$ should meet the following requirements:

$$\begin{array}{c}\{\begin{array}{c}{\theta }_{1\min }=0\hfill \\ {\theta }_{1\max }=\pi -\arccos \left(\frac{2\cos {\theta }_2+1}{\sqrt{5+4\cos {\theta }_2}}\right)-\arccos \left(\frac{\sqrt{5+4\cos {\theta }_2}}{2}\right)\hfill \end{array}\hfill \end{array}$$

(26)

Optimization algorithm

After the joint trajectories were described using FFS and the ranges of driving joints were also established, the last work was to optimize the number of items and the coefficients of the FFS with the goal of minimum energy consumption. However, this optimization objective function is nonlinear with a large number of variables. It is difficult to be solved by traditional multidimensional optimization methods, such as penalty function method, sequential quadratic programming, gradient descent method, and so on. Thus, an intelligent optimization algorithm, or rather, the genetic algorithm (GA) was used to plan the joint trajectories of the closed five-bow-shaped bar linkage, due to its good robustness and stability in solving high-dimensional nonlinear problems.^31,32

The method of GA can solve complex optimization problems in a relatively short computing time by using natural selection, natural genetic mechanisms in biology, and the principle of survival of the fittest. Optimization solution of GA is realized by the operations of selection operator, crossover operator, and mutation operator after encoding the variables as genes. Then, the coding mode, population size, fitness function, crossover probability, and mutation probability will impact the optimization solution on the accuracy and efficiency. The main steps of GA are as follows:

• (1) Coding and initializing population: Randomly generate a set of variables in N rows and K columns that satisfy the constraint conditions, where N is the number of individuals in the population and K is the number of genes that make up a chromosome. Each chromosome represents an individual in the population, and subsequent selection, crossover, mutation, and other operations are based on it.

• (2) Fitness function calculation: The initial population undergoes continuous iteration and elimination in GA, and better individuals need to be retained to ultimately obtain the optimal solution. The evaluation of individuals is based on the fitness value. In this paper, the fitness function is the energy consumption of the closed five-bow-shaped bar linkage.

• (3) Selection, crossover, and mutation: Individuals need to be selected based on their fitness. Selection refers to retaining excellent individuals in the population to form the next generation population. Crossing refers to the crossing of genes between individuals, forming two completely new individuals. Mutation is a random operation that generates new individuals by changing the genetic of certain individuals, in order to increase the diversity of the population and improve the search ability of the algorithm.

The parameters required for GA in MATLAB are shown in Table 1.

Table 1.

GA parameters in MATLAB.

Parameter	Meaning	Parameter	Meaning
Fitnessfcn	Fitness function	Nvars	Number of variables
X	Linear inequality coefficient matrix	Y	Linear inequality vector
I	Linear equation coefficient matrix	J	Linear equation vector
U	Minimum value vector	L	Maximum value vector

The fitness function is shown in equation (22), where the variables are the coefficients in the finite term Fourier series in equation (16), and the number of variables is $2\times M+1$ . According to the constraints on the driving joint in equations (23)–(26), the linear inequality constraint parameters X and Y can be obtained. Due to the lack of linear equality constraints and variable ranges, $I$ , $J$ , $U$ , $L$ are the default values.

Numerical examples

Example description

As the mentioned above, the rolling stage when $\varphi \in [0,2\mathrm{\pi }/5]$ could be only considered in the joint trajectories planning. The trajectory planning for other stages can refer to this example, due to the symmetrical design of the closed five-bow-shaped bar linkage. And then, we planned the joint trajectories for $\varphi \in [0,2\mathrm{\pi }/5]$ , with the following assumptions:

• (1) At both the initial and final moments, the outer contour of the linkage is circular while the inner contour maintains a positive pentagon, and the initial speed is equal to zero.

• (2) The linkage is assumed to be a multi-rigid-body system, and there is no relative sliding between the linkage and the ground at the touchdown point.

• (3) The duration of a motion period is 2 s, that is, $T_0=2\mathrm{s}$ . Namely, it takes 2 s for the linkage rolls from $\varphi =0$ to $\varphi =2\mathrm{\pi }/5$ .

In Section “Dynamic model of the closed five-bow-shaped bar linkage,” the dynamic model of the closed five-bow-shaped bar linkage was analyzed, and the generalized coordinate vector was defined. Based on the relationship between the driving and passive joints, the shape and position of the mechanism can be described through the three coupled coordinates mentioned above. Solving forward kinematics based on kinematic model is simple, but planning the joint trajectory through inverse kinematics is more complex, and the resulting solution is not unique. Das and Mukherjee used an equivalent rolling model to derive expressions of joint angular displacement with respect to the rolling angle in their study of a rolling disc. ³³ Similarly, by utilizing the relationship between joint angular displacement and roll angle described in equation (8), inverse kinematics analysis of the linkage can be carried out, as well as the joint trajectory planning. So, the first requirement is to design a suitable motion law for the roll angle $\varphi$ , followed by driving joint trajectories planning based on the variable roll angle.

In order to analyze the joint trajectory planning effect under different roll angular accelerations, we designed two roll angular acceleration cases, as shown as equations (27) and (28).

Case 1: Sinusoidal angular acceleration.

$$\stackrel{··}{\varphi }(t)=\frac{\pi }{5}sin\left(\pi t+\frac{3\pi }{2}\right)+\frac{\pi }{5}$$

(27)

Case 2: Modified trapezoidal angular acceleration.

$$\begin{array}{c}\stackrel{··}{\varphi }(t)=\{\begin{array}{c}\frac{t}{h_1}-\frac{1}{2\pi }\sin \left(\frac{2\pi }{h_1}t\right)t_0\le t\le t_1\hfill \\ 1t_1<t\le t_2\hfill \\ \frac{(t_3-t)}{h_2}-\frac{1}{2\pi }\sin \left[\frac{2\pi }{h_2}(t_3-t)\right]t_2<t\le t_3\hfill \end{array}\hfill \end{array}$$

(28)

where, $t_0=0\mathrm{s},t_1=0.7434\mathrm{s},t_2=1.2567\mathrm{s},t_3=2\mathrm{s},h_1=h_2=0.7434$  s. The roll angular accelerations of the two cases defined by equations (27) and (28) are shown as Figure 5. It can be seen that the angular displacements of the rolling angle for the two cases are equal to $2\mathrm{\pi }/5$ , and the angular velocities are equal to $0.4\mathrm{\pi }\mathrm{rad}/\mathrm{s}$ at the final moment.

Figure 5.

The angular velocities and accelerations of the two cases.

The followed work after the design of roll angular acceleration is to optimize the joint trajectories using GA. The suitable parameters significantly affect the optimization solution on the convergence speed and final results. It is generally believed that increasing the evolutionary algebra and the number of individuals participating in evolution can obtain optimal values. But this will lead to a rapid increasement of computational complexity. For our optimization problem, the main parameters of GA were set as shown in Table 2. And the main parameters of the closed five-bow-shaped bar linkage are also shown in Table 3.

Table 2.

Main parameters of the genetic algorithm.

Coding method	Population size	Crossover probability	Mutation probability	Iterations
Real encoding	300	0.9	0.03	500

Table 3.

Main parameters of the closed five-bow-shaped bar linkage.

Parameter	Values
Each bar mass ( $m_i$ )	1 kg
Moment of inertia $(J_i)$	0.0026  $\mathrm{kg}·{\mathrm{m}}^2$
Radius of curvature circle ( $R$ )	0.15 m
Chord length ( $l$ )	l = 0.016 m

Analysis of calculation results

The rolling acceleration laws described by equations (27) and (28) were respectively used to plan the joint trajectories which will be denoted by FFS with different number of items. Then, GA will be used to solve the coefficients of FFS with the goal of minimum energy consumption.

In order to analyze the optimization effect of the proposed trajectory planning method on energy consumption of the linkage, trajectories obtained by using constant potential energy rolling strategy in operating space in the study of Yu et al., ³⁴ were selected as comparison objects. In the work of Yu et al., a trajectory planning method that maintains a constant height of the center of mass was proposed, and the energy consumption was calculated using numerical methods. Its correctness was verified through ADAMS simulation. With the constant potential energy rolling strategy, energy consumptions of the linkage for the two acceleration cases are 0.217 J (Case 1) and 0.228 J (Case 2), respectively.

Energy consumption analysis

The number of items of the FFS were set as integers with the range of $2\le M\le 20$ , in order to compare the energy consumption of the linkage with different joint trajectories. The convergence processes of energy consumption optimization by GA for Case 1 (Sinusoidal angular acceleration) and Case 2 (modified trapezoidal angular acceleration) are illustrated in Figure 6. And the energy consumption trends over the number of items for the two roll angular acceleration cases are shown as Figure 7.

Figure 6.

Convergence process of GA.

Figure 7.

Energy consumption of the two rolling cases.

As shown as Figure 6, it can be seen that the energy consumption is convergent with an optimized result of $Q=0.171\mathrm{J}$ , when the iteration generation is approximately equal to 300. The GA is proved to be effective for our optimization problem. Furthermore, according to the energy consumption curves shown in Figure 7, we can obtain the following conclusions:

• (1) The trajectory planning method in this paper can effectively reduce energy consumption during the linear rolling motion of the closed five-bow-shaped bar linkage. In both Case 1 and Case 2, the energy consumption corresponding to trajectories with different numbers of items varies between 0.171 and 0.172 J. Comparing with the trajectories planned with constant potential energy strategy, the energy consumption was reduced by an average of 19% under the two different acceleration schemes. In addition, if we intercepted the same number of terms of FFS to fit the joint trajectories, the energy consumption of the linkage rolling with the sinusoidal angular acceleration (Described by equation (27)) would be lower than that of the modified trapezoidal angular acceleration (Described by equation (28)).

• (2) The number of terms intercepted from Fourier series has a great impact on the energy consumption. Namely, the energy consumption will decrease firstly and then increase with the increasement of $M$ , and the trajectory with the lowest energy consumption appears when $M\in [6,8]$ . Reason for this phenomenon is that the search space of GA will become larger with the increasement of $M$ and then makes the optimization effect weaken gradually.

In order to analyze the relationship between joint trajectory and the energy consumption of the linkage, the planned optimal trajectories of the two rolling angular acceleration cases were compared with the constant potential energy trajectories. ³⁴ And the comparing results are shown as Figures 8 and 9. It can be seen from the joint angular displacement curves shown in Figure 8, the angular displacements of the two driving joints ${\theta }_1$ and ${\theta }_2$ acquired by energy consumption optimization display an obvious symmetry. Comparing with the method based on constant potential energy, the joint trajectory planning method based on FFS with the goal of minimal energy consumption can equally distribute the motion laws of the rolling angle to both of the two driving joints, resulting in a lower energy consumption and a smaller peak in the joint trajectories. In addition, this rule of symmetrical distribution is also existed in the angular velocity curves of the two driving joints, as shown as Figure 9. The peak of the joint angular velocity planned in this paper is smaller while compared with the method based on constant potential energy. It demonstrated that decreasing the peak angular displacement as well as peak angular velocity is beneficial for energy reduction.

Figure 8.

Angular displacement of driving joints: (a) Case 1 and (b) Case 2.

Figure 9.

Angular velocity of driving joints: (a) Case 1 and (b) Case 2.

Movement analysis

At any moment of the rolling motion, the shape of the closed five-bow-shaped bar linkage is determined by the position of the system’s center of mass. The longitudinal offset (denoted as $y_s$ ) between the center of mass and the center of the virtual circle will determine the potential energy of the system, while the lateral offset (denoted as $x_s$ ) between them will affect the rolling acceleration. Since the trajectory of the center of mass has a great impact on the energy consumption of the system, it is necessary to compare its motion under the conditions of different terms of Fourier series. The centroid trajectories of the system obtained by intercepting the terms as $2\le M\le 20$ are shown as Figure 10. It can be seen that the centroid trajectories with different terms rise sharply first and then decrease slowly, both for Case 1 and Case 2. With the increase of number of terms ( $M\ge 4$ ), the centroid trajectories are gradually unified by same variation trend.

Figure 10.

Centroid trajectories for (a) Case 1 and (b) Case 2.

The number of terms in Fourier series intercepted to indicate the joint trajectory will have an impact on the motion of linkage. In order to analyze the influence of the number of terms, angular accelerations of the two driving joints for Case 2 were taken as an example to be shown as Figure 11. The angular accelerations when $M=3$ is shown as Figure 11(a). It can be seen from Figure 3(a), when a few of terms are intercepted, the joint trajectory expressed by FFS will be composed of a few trigonometric functions, resulting in an inadequate resolution to describe the motion characteristics of the joint trajectory. The joint angular acceleration curves tend to be smooth and stable while $M\in [6,8]$ , as shown as Figure 11(b). And the energy consumption of the linkage also appears within this range. As the number of terms increase, the joint trajectories contained some higher harmonics which showed high-frequency oscillations in small scales in the trajectories, as shown as Figure 11(c) and (d). Comparing Figure 11(c) with (d), the frequency of the fluctuations increases while the amplitude decreases as the number of terms continued to increase.

Figure 11.

Angular acceleration of the two driving joints for Case 2: (a) M = 3, (b) M = 7, (c) M = 11, and (d) M = 18.

Overall, the proposed trajectory planning method can effectively reduce energy consumption during the rolling process of the closed five-bow-shaped bar linkage. And moreover, an appropriate number of terms of Fourier series used to indicate the joint trajectory is of great importance for the properties of the linkage. Small number of terms is incapable of expressing the joint trajectories accurately, while large number of terms makes the joints to be subjected to vibration shock. Through the contrastive analysis, we can draw a conclusion that the joint trajectories denoted by FFS with a number of terms $6\le M\le 8$ are appropriate.

Simulation and experiments

In order to verify the effectiveness of the proposed trajectory planning method, the correctness of the linkage’s dynamic model and also the related theoretical conclusions, we conducted virtual prototype simulation experiments in ADAMS before the manufacturing of physical prototype. Inconsequential parts were eliminated from the virtual prototype, including the drive modules, control modules, and sensors. As a result, the virtual prototype has a minimum number of parts, that is, it composed of five bow-shaped bars only. Due to the gravitational bias torque causing the mechanism to roll, the coordinates of the center of mass determine its motion state. As mentioned above, the center of mass of each module should be located at the midpoint of the joint line, and the system center of mass should be at the center of the contour circle at initial and final moments. Through the analysis of the rod and system center of mass in Figure 12, it can be seen that the position of the center of mass is consistent with the mathematical model, proving the feasibility of the simplified model.

Figure 12.

Simplified model.

The parameters of each bar were set as same as those in the theoretical calculations. Contact relationship between the linkage and the ground was added at the outer contour of grounded bar. And the rotating pairs were added at each joint, where viscous frictions with the coefficients of $f_i=0.7\mathrm{N}·\mathrm{m}/\mathrm{s}(i=1,2,\dots ,5)$ also were added at the center of each joint. Then, the simplified virtual prototype was shown as Figure 13.

Figure 13.

Virtual prototype established in ADAMS.

The trajectories of driving joints acquired by our optimization method were selected as the inputs of the virtual prototype, correspondingly. And the angular displacement curves of the driving joints for Case 1 (obtained when $M=7$ ) and Case 2 (obtained when $M=6$ ) are shown in Figure 14.

Figure 14.

Inputs of the two driving joints: (a) Case 1 and (b) Case 2.

The theoretical calculation and the simulation results for Case 1 and Case 2 were shown as Figures 15 and 16, respectively. From Figures 15(a) and 16(a), it can be seen that the simulation roll angles for Case 1 and Case 2 both have the same variation trend with the theoretical results, but have a time lag phenomenon. This time lag can be explained by the lack of consideration of friction between the touchdown bar and the ground in the theoretical calculation. From Figures 15(b) and 16(b), it also can be seen that the simulation driving torques for Case 1 and Case 2 both have the same variation trend with the theoretical results, but have small relative errors between them due to the insufficient accuracy of the virtual prototype. Using the numerical method in Section “Planning objective,” the system energy consumption during the simulation can be obtained based on equation (22). The simulated energy consumption in Case 1 and Case 2 were 0.182 and 0.185 J, respectively, slightly higher than the calculated result. In general, the dynamic model, trajectory planning method, and also the energy consumption variation pattern of the closed five-bow-shaped bar linkage were verified by the simulation results.

Figure 15.

Calculation and simulation results of Case 1: (a) roll angle and (b) driving torque.

Figure 16.

Calculation and simulation results of Case 2: (a) roll angle and (b) driving torque.

After the verification on virtual prototype, a simple physical prototype was also made by 3D printing using resin. And digital actuators with the PWM pulse width control mode were used to drive the joints. Then, an experimental system which includes a control unit, power supply, linkage system, and servo motors was set up to examine the results. The experimental system and its mechatronic system are respectively shown as Figures 17 and 18.

Figure 17.

Experimental system.

Figure 18.

Mechatronic system.

The experimental system uses Arduino Uno as the control core, and Arduino is a completely open source electronic platform that can be modified and designed freely under its open source protocol. At the same time, this control board has relatively flexible software and hardware, which can provide more I/O interfaces, connect more sensor modules, and ensure the stability and motion accuracy of the closed five-bow-shaped bar linkage. In the experiment, LDX-227 motor was used to control the driving joint, which can achieve rotation from $0°$ to $270°$ and output a maximum torque of $15\mathrm{kg}·\mathrm{cm}$ . The servo motors at the two driving joints are supplied with 7.5 V voltage after battery voltage reduction, and the Arduino UNO control board is connected to an external 5 V power supply to ensure stable operation. The GND terminals of the motor are connected to the GND terminals of the control board, and the PWM terminals of the motor are respectively connected to the corresponding PWM terminals on the control board. The control program is written using the Arduino IDE and uploaded to the control board. The control program contains finite Fourier series driving joint trajectories and time arrangements. When the program starts, the control system reads the target angle of two driving joints every 0.2 s, calculates the corresponding PWM signal duty cycle, outputs two PWM control signals, drives the motors to rotate, and cause the mechanism to roll.

During the experiment, a camera was used to take pictures of the acceleration and rolling process of the closed five-bow-shaped bar linkage. The camera was set to take two photos per second. In order to compare the experimental results with the theoretical value, photos from the experiment were used to manually analyze rolling angles. The two driving joints were controlled using the trajectories shown in Figure 14(a), in view of the less energy consumption. The plots of the linear rolling motion of the closed five-bow-shaped bar linkage were demonstrated as Figure 19. The comparison between the rolling angle obtained from the experiment and the calculation is shown in Figure 20. It can be seen that the rolling speed of the mechanism in the experiment is faster than the theoretical results, which is due to the insufficient accuracy of the experimental prototype. Due to limitations in experimental equipment, the accuracy of the data is not sufficient for detailed comparative analysis of motion. In the subsequent work, accurate roll angle curves can be obtained by increasing the frequency of photography, and joint trajectory curves can be obtained by installing angle sensors at the joint flange and achieving feedback control loop design. As the rolling angle displacement could be accord with the planned values, it proves that the linkage can roll well with planned trajectories.

Figure 19.

Rolling motion test plots: (a) time = 0 s, (b) time = 0.5 s, (c) time = 1 s, (d) time = 1.5 s, and (e) time = 2 s.

Figure 20.

Roll angle paths.

Conclusions

This paper documented a study of trajectory planning method for a closed five-bow-shaped bar linkage whose joint trajectories were expressed by finite Fourier series. The mechanism is transformed from a planar five bar mechanism, and its dynamic model has been studied in detail. The Genetic algorithm was used to search the optimal energy consumption trajectory under several constraints. After the theoretical calculation, simulation on the virtual model and also experiments on a simple prototype, the following conclusions can be obtained:

• (1) Compared with commonly used trajectory planning curves, Fourier series has the advantages of concise expression and high-order differentiability. The joint trajectory in the whole region can be expressed by finite Fourier series, as it has the ability to represent any periodic functions. This characteristic makes the optimization algorithm especially GA can search joint trajectories in the whole feasible region with the target of energy consumption, thus, makes the linkage roll with the lowest energy consumption. Then, energy consumption of the linkage based on the trajectory planning method using FFS and GA can be reduced by approximately 19%, comparing with that of the trajectory planned with a constant potential energy strategy in operation space.

• (2) An appropriate number of items intercepted from Fourier series to denote joint trajectories is important for the linkage. Trajectories when intercepting 0–20 items are planned and analyzed. As a result, small number of terms is incapable of expressing the joint trajectories accurately, while large number of terms makes the joints to be subjected to vibration shock. The number of terms $M$ within the scope of $6\le M\le 8$ is suitable.

• (3) After simplifying the mechanical model and analyzing the control methods, simulation on a virtual model and also the prototype experiment verifies that the trajectory planning method based on Fourier series can design the motion of a closed five-bow-shaped bar linkage in the whole region.