A hierarchical approach for rigid-body dynamics model simplification of a high-speed parallel robot by considering kinematics performance

Abstract

Considering the real-time control of a high-speed parallel robot, a concise and precise dynamics model is essential for the design of the dynamics controller. However, the complete rigid-body dynamics model of parallel robots is too complex for online calculation. Therefore, a hierarchical approach for dynamics model simplification, which considers the kinematics performance, is proposed in this paper. Firstly, considering the motion smoothness of the end-effector, trajectory planning based on the workspace discretization is carried out. Then, the effects of the trajectory parameters and acceleration types on the trajectory planning are discussed. But for the fifth-order and seventh-order B-spline acceleration types, the trajectory will generate excessive deformation after trajectory planning. Therefore, a comprehensive index that considers both the motion smoothness and trajectory deformation is proposed. Finally, the dynamics model simplification method based on the combined mass distribution coefficients is studied. Results show that the hierarchical approach can guarantee both the excellent kinematics performance of the parallel robot and the accuracy of the simplified dynamics model under different trajectory parameters and acceleration types. Meanwhile, the method proposed in the paper can be applied to the design of the dynamics controller to enhance the robot's performance.

Introduction

Parallel robots have attracted extensive attention from academia and industry over the last few decades.^1–3 Due to the closed-loop structure features, compared with serial robots, parallel robots own advantages in stiffness and dynamics, which make them possible to work at a high speed and yet maintain a high level of accuracy. Therefore, parallel robots are widely used in industrial production lines to complete high-speed sorting, packing, stacking, and assembling. And this can be certificated by the successful applications of the well-known Delta robot and its modified versions.^4–6

Because parallel robots usually work at a high speed, the influences of residual vibration on the working accuracy cannot be ignored. In view of this, the control algorithm should consider both the kinematics and dynamics performance of parallel robots. Therefore, it is very important to study the dynamics modeling method that is adaptive to the design of the controller. For dynamics modeling, there are many methods that can be used, such as Newton–Euler formulation, ⁷ Lagrange formulation,^8,9 virtual work principle,^10,11 and some other methods. ¹² Among them, the virtual work principle can avoid the calculation of the internal force term, and the solving process is relatively simple. Therefore, it is considered one of the most effective methods for dynamics modeling. However, the complete dynamics model of a parallel robot based on the virtual work is not concise for real-time control. Therefore, it is necessary to simplify it properly. Considering this, many scholars have done a lot of research on the simplification method of the dynamics model. To propose a general and concise dynamics modeling method, Orsino et al.^13–15 proposed a modular modeling methodology for parallel robots, where any restriction concerning either the space index or the number of degrees of freedom does not need to be considered. Considering the contribution of different torque components, Carbonari ¹⁶ proposed a polynomial simplified dynamics model for the 3-central processing unit class of parallel robots by fitting the actual behavior of the robot prototype, which shows substantial reliability of the model in non-static conditions and advantages in terms of computation time. Considering the motion ranges of the legs are smaller than those of the moving platform and the sliders, Gao et al. ¹⁷ proposed a simplified modeling method for the 3-PRRR/PPRR (P-Prismatic, R-Revolute) redundantly actuated parallel mechanism, where the mass of the legs are divided into two parts and concentrated in the moving platform and the sliders, respectively. Without considering the rotational inertia of the passive limbs, Codourey ¹⁸ proposed a dynamics model simplification method of the Delta parallel robot by dividing the mass of the passive limbs into two parts, where two-thirds are attached to the end of the active limbs and the rest is attached to the moving platform. Based on the calculation principle of the rotational inertia, Wu et al. ¹⁹ proposed a similar model simplification method, where the mass of the passive limbs is equally distributed to the end of the active limbs and the moving platform. Subsequently, many scholars analyzed and demonstrated the simplified criteria based on the above-mentioned mass distribution method.^20–23

It should be noted that there is a lot of researches on the dynamics model simplification, while the influence of kinematics characteristics on the model simplification is seldom studied. In fact, the dynamics performance of the robot can be affected by kinematics characteristics. Although many scholars have conducted extensive research on trajectory planning to improve kinematics performance, seldom combine it with dynamics model simplification. And the dynamics characteristic is different under different trajectory parameters and acceleration types, so the simplified dynamics model adopted by the single fixed coefficient will generate a large error. Therefore, a dynamics simplification method based on the combined coefficients, which considers kinematics performance, is proposed in this paper. And based on this, taking the Delta robot as an example, the integrated analysis of trajectory planning and dynamics model simplification under different trajectory parameters and acceleration types is studied. In brief, considering kinematics performance, a hierarchical simplification method of rigid-body dynamics modeling based on combination coefficients is proposed in this paper, which can be implemented in the design of the dynamics controller to enhance the robot performance.

In this work, the rest of this paper is organized as follows: section “Kinematics and dynamics” establishes the kinematics and dynamics model of the robot. And a hierarchical approach for dynamics model simplification based on the combined coefficients is proposed for subsequent optimization analysis. In section “Trajectory planning,” the trajectory planning based on the workspace discretization is studied. After trajectory planning, the dynamics model simplification method based on the combined mass distribution coefficients is introduced in section “Dynamics model simplification.”. Finally, the main conclusions are drawn in section “Conclusions.”

Kinematics and dynamics analysis

As shown in Figure 1, the Delta robot is composed of a static platform, a moving platform, and three identical kinematic chains, which are connected to the static platform and moving platform by revolute joints and spherical joints, respectively. Each kinematic chain contains an active limb and a passive limb of a parallelogram structure, which are connected to each other by a spherical joint. Motors and reducers are installed on the static platform to reduce the mass of moving parts. Driven independently by three servo motors, the robot provides the moving platform with a 3-degree of freedom translational motion capability.

Figure 1.

The computer-aided design (CAD) model of the Delta robot.

Kinematics analysis

For the convenience of analysis, a schematic diagram of the Delta parallel robot is depicted in Figure 2. The global coordinate system O–xyz is attached to the geometric center of the static platform, with the x-axis pointing to point A₁, the y-axis parallel to A₃A₂, and the z-axis can be determined by the right-hand rule. Since there is no rotation of the moving platform, the moving platform can be simplified to a point P. A_i is the center of the ith revolute joint. The passive limb is a quadrilateral mechanism, thus it can be represented by a limb B_iP that goes through the midpoint of two short sides.

Figure 2.

The schematic diagram of the Delta robot.

In the global coordinate system, the position vector of the manipulator endpoint P can be expressed as

$${\mathit{r}}_p={\mathit{e}}_i+l_1{\mathit{u}}_i+l_2{\mathit{w}}_i(i=1,2,3)$$

(1)

where ${\mathit{e}}_i=e{\left(\begin{array}{ccc}\cos {\beta }_i& \sin {\beta }_i& 0\end{array}\right)}^T$ , e is the radius difference between the moving platform and static platform. ${\beta }_i$ is the joint angle of the static platform, ${\beta }_i=(i-1)(2\pi /3)$ . $l_1$ , $l_2$ , ${\mathit{u}}_i$ , and ${\mathit{w}}_i$ are the length and unit vector of an active limb and a passive limb, respectively. ${\mathit{u}}_i=(\cos {\beta }_i\cos {\theta }_i\sin {\beta }_i\cos {\theta }_i-\sin {\theta }_i{)}^T$ , ${\theta }_i$ is the rotation angle of the ith active limb.

Through the assembly relationship of the Delta robot, the inverse position solution model can be formulated as

$${\theta }_{pi}=2\arctan {\displaystyle \frac{-E_i-\sqrt{E_i^2-G_i^2+F_i^2}}{G_i-F_i}}$$

(2)

where

$$\begin{array}{rl}& E_i=2l_1({\mathit{r}}_p-{\mathit{e}}_i{)}^T\hat{\mathit{z}}\\ & F_i=-2l_1({\mathit{r}}_p-{\mathit{e}}_i{)}^T(\cos {\beta }_i\hat{\mathit{x}}+\sin {\beta }_i\hat{\mathit{y}})\\ & G_i=({\mathit{r}}_p-{\mathit{e}}_i{)}^T({\mathit{r}}_p-{\mathit{e}}_i)+l_1^2-l_2^2\\ & \hat{x}=(1,0,0{)}^T,\hat{y}=(0,1,0{)}^T,\hat{z}=(0,0,1{)}^{\mathrm{T}}\end{array}$$

According to equation (1), the unit vector of a passive limb can be obtained

$${\mathit{w}}_i={\displaystyle \frac{1}{l_2}({\mathit{r}}_p-{\mathit{e}}_i-l_1{\mathit{u}}_i)}$$

(3)

Taking the derivative of equation (1) with respect to time, the angular velocity vector of active limbs can be expressed as

$${\mathit{\theta }}_v=\mathit{J}{\mathit{r}}_v,\begin{array}{c}\end{array}{r}_v=[v_x,v_y,v_z{]}^{\mathrm{T}}$$

(4)

where

$$\mathit{J}={\left[\begin{array}{ccc}{l}_1{\displaystyle \frac{{\mathit{w}}_1^{\mathrm{T}}}{{\mathit{w}}_1^{\mathrm{T}}({\mathit{v}}_1\times {\mathit{u}}_1)}}& l_1{\displaystyle \frac{{\mathit{w}}_2^{\mathrm{T}}}{{\mathit{w}}_2^{\mathrm{T}}({\mathit{v}}_2\times {\mathit{u}}_2)}}& l_1{\displaystyle \frac{{\mathit{w}}_3^{\mathrm{T}}}{{\mathit{w}}_3^{\mathrm{T}}({\mathit{v}}_3\times {\mathit{u}}_3)}}\end{array}\right]}^{\mathrm{T}}$$

In the same way, the angular velocity vector of a passive limb can also be solved as

$${\mathit{\omega }}_{vi}={\mathit{J}}_{wi}{\mathit{r}}_v$$

(5)

where

$${\mathit{J}}_{wi}={\displaystyle \frac{1}{l_2}([{\mathit{w}}_i\times ]-l_1{\mathit{w}}_i\times ({\mathit{v}}_i\times {\mathit{u}}_i){\mathit{J}}_i)}$$

Taking the quadratic derivative of equation (1) with respect to time, the angular acceleration vector of active limbs can be derived as

$${\mathit{\theta }}_a=\mathit{J}{\mathit{r}}_a+F({\mathit{r}}_v),\begin{array}{c}\end{array}{r}_a=[a_x,a_y,a_z{]}^{\mathrm{T}}$$

(6)

where ${\mathit{\theta }}_a$ is the angular acceleration vector of active limbs, ${\mathit{r}}_a$ is the acceleration vector of the moving platform, and $F({\mathit{r}}_v)$ is the non-linear velocity term.

Through the same way, the acceleration vector of a passive limb can also be derived as

$${\mathit{\omega }}_{ai}={\mathit{J}}_{wi}{\mathit{r}}_a+K_i(r_v)$$

(7)

where ${\mathit{\omega }}_{ai}$ is the angular acceleration vector of the passive limb and $K_i(r_v)$ is the non-linear velocity term.

Combining equations (6) and (7), the centroid acceleration vector of the passive limb can be expressed as

$${\mathit{r}}_{ca,i}={\mathit{J}}_{vi}{\mathit{r}}_a+N_i(r_v)$$

(8)

where ${\mathit{r}}_{ca,i}$ is the centroid acceleration vector of a passive limb and $N_i(r_v)$ is the non-linear velocity term.

Taking the third derivative of equation (1) with respect to time, the angular jerk vector of active limbs can be derived as

$${\mathit{\theta }}_j=\mathit{J}{\mathit{r}}_j+R({\mathit{r}}_v,{\mathit{r}}_a),\begin{array}{c}\end{array}{r}_j=[j_x,j_y,j_z{]}^{\mathrm{T}}$$

(9)

where ${\mathit{\theta }}_j$ is the angular jerk vector of active limbs, $r_j$ is the jerk vector of the moving platform, and $R({\mathit{r}}_v,{\mathit{r}}_a)$ is the non-linear velocity and acceleration term.

Dynamics analysis

For the convenience of analysis, the local coordinate O–x_iy_iz_i is attached to the centroid of the ith passive limb, and the z-axis is along the axis of the passive limb, the x-axis is perpendicular to the z-axis and in the plane composed of $w_i$ and $w_i\times u_i$ , the y-axis can be determined by the right-hand rule. Therefore, ignoring the friction, the complete rigid-body dynamics model based on the virtual work principle^24–26 can be written as

$${\mathit{\tau }}_{\mathrm{com}}={\mathit{\tau }}_{\mathrm{com},a}+{\mathit{\tau }}_{\mathrm{com},v}+{\mathit{\tau }}_{\mathrm{com},g}+{\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}$$

(10)

where

$$\begin{array}{rl}& {\mathit{\tau }}_{\mathrm{com},a}=(I_{\mathrm{A}}\mathit{J}+{\mathit{J}}^{-\mathrm{T}}m){\mathit{r}}_a\\ & {\mathit{\tau }}_{\mathrm{com},v}=I_{\mathrm{A}}f(r_v)\\ & {\mathit{\tau }}_{\mathrm{com},g}=m_{\mathrm{A}}{r}_{A}g{\left(\begin{array}{ccc}\cos {\theta }_1& \cos {\theta }_2& \cos {\theta }_3\end{array}\right)}^{\mathrm{T}}+{\mathit{J}}^{-\mathrm{T}}mg\hat{\mathit{z}}\\ & {\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}=J^{-\mathrm{T}}\sum _{i=1}^3(J_{wi}^{\mathrm{T}}{M}_{\mathrm{rod}i}+J_{vi}^T{f}_{si})\end{array}$$

According to the complete rigid-body dynamics model, the input torque of the joints ${\mathit{\tau }}_{\mathrm{com}}$ is composed of four parts. ${\mathit{\tau }}_{\mathrm{com},a}$ is the acceleration torque term that depends on both configuration and acceleration of the robot, ${\mathit{\tau }}_{\mathrm{com},v}$ is the velocity torque term that depends on the velocity of the robot, ${\mathit{\tau }}_{\mathrm{com},g}$ is the gravity torque term that depends on the configuration of the robot, ${\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}$ is the passive limb torque. $I_A$ is the equivalent moment of inertia of the active limb about its axis of rotation, $m_A{r}_A$ is the mass–radius product of the active limb assembly, and m is the equivalent mass of the moving platform.

Simplification strategy analysis

To establish a simplified dynamics model based on the combined mass distribution coefficients, which considers the kinematics performance of the robot, a hierarchical procedure is proposed in this section. As shown in Figure 3, firstly, an acceleration type is selected initially according to the experiences. Secondly, the trajectory planning based on the workspace discretization is carried out. Then, based on the optimal results, analyzing the output performance of the robot whether can meet the mission requirements. When the terminating condition (TC) is satisfied, the determination of the combination coefficients is carried out in Step 3. Finally, the final optimal results can be obtained.

Figure 3.

The hierarchical procedure of dynamics model simplification.

Trajectory planning

Description of a typical trajectory

As the Delta parallel robot is mainly used to complete the grasping and placing operations in practical engineering, the typical motion trajectory of the end-effector in the workspace is a “door” type as shown in Figure 4, which includes some key points P₁–P₇. The vertical height and horizontal width of the trajectory are h_a and b_a, respectively. The h_m is the vertical distance from P₁ to P₂. And $\varphi$ is the orientation angle of the trajectory relative to the local coordinate system. For the convenience of analysis, assume that the time of ascent and descent movement is T₁ and the time of horizontal movement is T₂. In addition, this paper adopts a stand motion trajectory where h_a=0.7 m, b_a=0.025 m, and h_m=h_a/2 for the subsequent analysis. The workspace is a cylindrical space with the radius R and the height h. The O–x_wy_wz_w is the local coordinate system that is obtained by translating the global coordinate a distance of H + h − h_d along the negative direction of the z-axis, where the H is the vertical distance from the upper surface of the workspace to the static platform, and h_d is the vertical height of the lowest point of trajectory relative to the bottom surface of the workspace.

Figure 4.

The schematic of the motion trajectory in the workspace.

Selection of acceleration types

3–4–5 polynomial

In terms of the selection of acceleration types, there are many acceleration types that can be adopted in trajectory planning, such as polynomial acceleration law, trapezoidal acceleration law, sine acceleration law, and B-spline acceleration law. Among them, the 3–4–5 polynomial is the most commonly used and effective acceleration type, which can be expressed as

$$\begin{array}{rl}& s(t)=10S(t^3/T^3)-15S(t^4/T^4)+6S(t^5/T^5)\\ & v(t)=30S(t^2/T^3)-60S(t^3/T^4)+30S(t^4/T^5)\\ & a(t)=60S(t/T^3)-180S(t^2/T^4)+120S(t^3/T^5)\\ & j(t)=60S(1/T^3)-360S(t/T^4)+360S(t^2/T^5)\end{array}$$

(11)

where t is the motion time, T is the motion period, and S is the length of a typical path.

B-spline

Based on the B-spline acceleration law, trajectory planning can make the curves of velocity, acceleration, and even jerk continuously. Meanwhile, the values of velocity, acceleration, and jerk at the starting and ending points can be arbitrarily defined. A B-spline curve of order k is a piecewise polynomial, which can be formulated as

$$p(u)=\sum _{i=0}^n{d}_i{N}_{i,k}(u)$$

(12)

where $d_i$ is the control point, $N_{i,k}(u)$ is the ith Bernstein basis function derived by the De Boor formula, which can be expressed as

$$\{\begin{array}{c}{N}_{i,0}(u)=\{\begin{array}{c}\begin{array}{cc}1,& u_i\le u\le u_{i+1}\end{array}\\ \begin{array}{cc}0,& \mathrm{else}\begin{array}{ccc}& & \end{array}\end{array}\end{array}\begin{array}{ccc}& & \begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& & \end{array}\end{array}\\ N_{i,k}(u)={\displaystyle \frac{u-u_i}{u_{i+k}-u_i}{N}_{i,k-1}(u)+{\displaystyle \frac{u_{i+k+1}-u}{u_{i+k+1}-u_{i+1}}{N}_{i+1,k-1}(u)}}\end{array}$$

(13)

where u_i is the time node of the B-spline of order k, which can be expressed as

$$u_i=\{\begin{array}{c}0,i=0,\dots ,k\\ {u_i}_{-1}+|\mathrm{\Delta }{t}_{i-k-1}|/T,i=k+1,\dots ,n+k-1\\ 1,i=n+k,\dots ,n+2k\end{array}$$

(14)

where T is the motion period of a typical trajectory.

Accordingly, the rth derivative of $p(u)$ with respect to u can be expressed as

$$p^r(u)={\sum }_{j=i-k+r}^i{d}_j^r{N}_{j,k}(u)$$

(15)

where $d_j^r$ is the rth derivative of the control point and can be obtained recursively as

$$d_j^r=\{\begin{array}{c}{d}_j,r=0\begin{array}{ccc}& & \begin{array}{cc}& \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& \end{array}\end{array}\end{array}\\ (k+1-r){\displaystyle \frac{d_j^{i-1}-d_{j-1}^{i-1}}{u_{j+k}-u_j},r=1,2,\dots ,k}\end{array}$$

(16)

Considering the boundary conditions in terms of velocity, acceleration, and jerk at both the start point and endpoint of the trajectory, the following constraint equations are obtained:

$$\begin{array}{rl}& p^1(u_k)=d_1^1,p^1(u_{n+k})=d_{n+k+1}^1,p^2(u_k)=d_1^2\\ & \\ & p^2(u_{n+k})=d_{n+k+1}^2,p^3(u_k)=d_1^3,p^3(u_{n+k})=d_{n+k+1}^3\end{array}$$

(17)

Therefore, control points of the B-spline can be solved

$$d={\mathit{C}}^{-1}p$$

(18)

Comparison of different acceleration types

For the convenience of analysis, taking a typical case where the trajectory parameter h_d = 0.125 m, $\varphi =0^{\mathrm{o}}$ and the segment time T₁ = 0.1 s, T₂ = 0.15 s as an example, the trajectory of the end-effector, the angular velocity, the angular acceleration, and the input joint torque of the first joint are depicted in Figure 5, which are defined as output performance of the robot. As can be observed, the output performances of the robot based on the 3–4–5 polynomial acceleration law and B-spline acceleration laws are different. For details, in terms of the output trajectory that is based on the B-spline acceleration law produces slight deformation while that based on the 3–4–5 polynomial acceleration law produces almost no deformation. In terms of angular velocity that is based on the 3–4–5 polynomial acceleration law and B-spline acceleration laws are almost the same, but for the angular acceleration and torque, there are several sharp inflection points in the curves that are based on the 3–4–5 polynomial acceleration law, which may cause shocks in the process of movement. Comparing the torque based on the 3–4–5 polynomial acceleration motion law and B-spline acceleration motion laws in Figure 5(d), it can be found that adopting the fifth-order and seventh-order B-spline acceleration laws in the trajectory planning is effective to reduce the required torque of servomotors. In particular, for the seventh-order B-spline, the angular acceleration and the torque at the starting and ending stages are smaller, but for the fifth-order B-spline, the peak torque and the peak angular acceleration are smaller. In brief, for the same trajectory parameters and segment time, the output performance based on different acceleration types is different. Therefore, the selection of acceleration types should be determined according to the actual application conditions.

Figure 5.

The output performance comparison between 3–4–5 polynomial and B-splines.

Trajectory optimization

As shown in Figure 5, when the different acceleration types are adopted, the output performances of the robot under the same trajectory parameters and segment time are different. Therefore, it is necessary to analyze trajectory planning under different acceleration types. Since the Delta parallel robot is mainly used to complete the grasping and placing operations in practical engineering, the motion smoothness of the end-effector is very important. Therefore, the trajectory planning based on motion smoothness will be conducted in this section. In addition, the trajectory parameters such as $h_d$ and $\varphi$ can also affect the motion smoothness of the end-effector. To analyze the effects of trajectory parameters on trajectory planning, the workspace is dispersed by the way shown in Figure 6. Since there are three identical kinematic chains in the Delta robot, the workspace of the robot is divided into six identical regions. Therefore, this study takes $0^o\le \varphi \le {60}^o$ as an example, and the height and angle discrete spacing are $\mathrm{\Delta }h$ and $\mathrm{\Delta }\varphi$ , respectively.

Figure 6.

The discretization of workspace.

As the motion smoothness of the trajectory is mainly reflected in the value of the jerk. It is necessary to study trajectory planning based on the minimum jerk. Therefore, considering the average cumulative effects of the joint jerk in a single movement period, the following index can be defined to evaluate the motion smoothness of the trajectory:

$$K_{\mathrm{jerk}}=\sqrt{{\displaystyle \frac{(1/3)\sum _{i=1}^3{\int }_0^{T_{\mathrm{total}}}{(j_i)}^2\mathrm{d}t}{{\int }_0^{T_{\mathrm{total}}}\mathrm{d}t}}}$$

(19)

where T_total is the movement period. So the trajectory optimization model for different acceleration types based on the workspace discretization can be shown as follows:

$$\begin{array}{rl}& \begin{array}{ccc}& & \begin{array}{ccc}& & \begin{array}{ccc}\begin{array}{cc}& \end{array}& \begin{array}{cc}& \end{array}& min\end{array}\end{array}\end{array}{K}_{\mathrm{jerk}}\\ & \begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& & \end{array}\begin{array}{cc}\mathrm{s}.\mathrm{t}.& \begin{array}{cc}& \end{array}\end{array}{T}_1+T_2=T_{\mathrm{total}}\\ & \begin{array}{ccc}\begin{array}{cc}& \end{array}& & \end{array}{\theta }_{min}-\underset{i\in [1,3]}{min}({\theta }_i)\le 0,\underset{i\in [1,3]}{max}({\theta }_i)-{\theta }_{max}\le 0\\ & \underset{i\in [1,3]}{max}(|v_i|)-v_{Mi,max}\le 0,\underset{i\in [1,3]}{max}(|{\tau }_i|)-{\tau }_{Mi,max}\le 0,\underset{i\in [1,3]}{max}(P_i)-P_{Mi,max}\le 0\end{array}$$

(20)

where ${\theta }_{min}$ and ${\theta }_{max}$ are the minimum and maximum value of the robot joint rotation angle, respectively, $v_{Mi,max}$ , ${\tau }_{Mi,max}$ , and $P_{Mi,max}$ are the maximum value of the velocity, torque, and power of the servo motor, respectively.

Based on the minimum jerk optimization, the final optimization results are shown in Figure 7. It can be seen that the trajectory parameters $h_d$ and ${\varphi }_{}^{}$ have a great influence on the segment time $T_1$ and the jerk index $K_{\mathrm{jerk}}$ . To be specific, the influence laws of the trajectory parameters on the optimization of segment time $T_1$ are different when different acceleration types are adopted. On the contrary, after the optimization, the effects of the trajectory parameters $h_d$ and ${\varphi }_{}^{}$ on $K_{\mathrm{jerk}}$ are roughly the same. Furthermore, $K_{\mathrm{jerk}}$ is mainly affected by $h_d$ and increases with the increase of $h_d$ .

Figure 7.

The results after minimum jerk optimization.

To analyze the above phenomenon, taking the 3–4–5 polynomial acceleration law as an example, the relationship between the angular motion range of active limbs $\mathrm{\Delta }\theta$ and $h_d$ (i.e. taking $\varphi =0^{\mathrm{o}}$ as an example), the relationship between the angular motion range of active limbs $\mathrm{\Delta }\theta$ and ${\varphi }_{}^{}$ (i.e. taking $h_d=0.125\mathrm{m}$ as an example) are depicted in Figure 8. As can be observed, for the same ${\varphi }_{}^{}$ , the mean angular motion range of active limbs $\mathrm{\Delta }{\theta }_{\mathrm{mean}}$ and the angular motion range of the ith active limb $\mathrm{\Delta }{\theta }_i$ all increase with $h_d$ . But for the same $h_d$ , $\mathrm{\Delta }{\theta }_{\mathrm{mean}}$ remains almost unchanged with the increase of ${\varphi }_{}^{}$ , and the change trend of $\mathrm{\Delta }{\theta }_i$ is different. In general, for a given trajectory and motion period, the smoothness of the motion is reflected by the average motion range of the active joints to a certain extent. So, for the same ${\varphi }_{}^{}$ , $K_{\mathrm{jerk}}$ increases with $h_d$ . Meanwhile, for the same $h_d$ , $K_{\mathrm{jerk}}$ remains almost unchanged with the increase of ${\varphi }_{}^{}$ . But for the optimization of the segment time $T_1$ based on the workspace discretization, the acceleration type and the motion of three joints should be considered at the same time. So the influences of the trajectory parameters on the optimization of the segment time $T_1$ are different when different acceleration types are adopted.

Figure 8.

The relationship between $\mathrm{\Delta }\theta$ and trajectory parameters.

To compare the motion parameters of robot joints before and after minimum jerk optimization, taking $h_d=0.125\mathrm{m}$ and $\varphi =0^{\circ }$ as an example, the optimization results are shown in Table 1. And it can see that the optimization method based on the minimum jerk can effectively reduce the maximum speed, maximum acceleration, maximum jerk, and maximum input torques of the robot joints. Furthermore, after the minimum jerk optimization, the maximum jerks of the 3–4–5 polynomial acceleration law and B-spline acceleration laws are 6.82 × 10⁵°/s³, 3.57 × 10⁵ °/s³, and 4.87 × 10⁵°/s³, respectively, which are 51%, 35%, and 36% lower than those of the typical trajectory. And comparing with the typical trajectory, the maximum input torques of robot joints are also reduced by 27%, 35%, and 26%, respectively. So the minimum jerk optimization model can reduce the input torques of robot joints and improve the smoothness of the trajectory.

Table 1.

The motion parameters of robot joints before and after minimum jerk optimization (units: s, s, °/s, °/s², °/s³, N·m).

	Acceleration type	T 1	T 2	max|v|	max|a|	max|j|	max|τ|
Before optimization	3–4–5 polynomial	0.100	0.150	8.28 × 102	1.92 × 104	1.38 × 106	15.77
	The fifth-order B-spline	0.100	0.150	6.75 × 102	1.35 × 104	5.50 × 105	10.45
	The seventh-order B-spline	0.100	0.150	7.25 × 102	1.53 × 104	7.58 × 105	11.87
After optimization	3–4–5 polynomial	0.073	0.177	7.01 × 102	1.47 × 104	6.82 × 105	11.49
	The fifth-order B-spline	0.035	0.215	4.93 × 102	9.91 × 103	3.57 × 105	6.78
	The seventh-order B-spline	0.050	0.200	6.00 × 102	1.28 × 104	4.87 × 105	8.73

However, adopting the optimized segment time in Table 1, for the fifth-order and seventh-order B-spline acceleration laws, the above optimization method can cause the deformations of the trajectory curve, which are 43.1 mm and 52.4 mm, as shown in Figure 9, so it is necessary to suppress the deformation during trajectory planning.

Figure 9.

The trajectory after minimum jerk optimization.

According to equation (18), once the must-pass point sequence p is given, when calculating the control vertexes d of the trajectory based on B-spline acceleration laws, the control matrix ${\mathit{C}}^{-1}$ is equivalent to producing the linear influence on the must-pass points. Therefore, equation (18) can be regarded as a system of linear equations. So the trajectory deformation problem can be regarded as the perturbation error analysis of a system of linear equations.

Firstly, taking the norm of equation (18), $||d||\le ||{\mathit{C}}^{-1}||\bullet ||p||$ can be obtained. And according to equation (18), p can be expressed as $p=\mathit{C}d$ . Assuming d has a small disturbance $\mathrm{\Delta }d$ , then p will generate a small disturbance $\mathrm{\Delta }p$ , which can be expressed as $\mathrm{\Delta }p=\mathit{C}\mathrm{\Delta }d$ , so $||\mathrm{\Delta }p||\le ||\mathit{C}||\bullet ||\mathrm{\Delta }d||$ can be obtained. Then the following inequality can be solved:

$${\displaystyle \frac{||\mathrm{\Delta }p||}{||p||}\le ||\mathit{C}||\bullet ||{\mathit{C}}^{-1}||{\displaystyle \frac{||\mathrm{\Delta }d||}{||d||}=\mathrm{cond}({\mathit{C}}^{-1}){\displaystyle \frac{||\mathrm{\Delta }d||}{||d||}}}}$$

(21)

Secondly, assuming the p has a small disturbance $\mathrm{\Delta }p$ , then the d will generate a small disturbance $\mathrm{\Delta }d$ , which can be expressed as $\mathrm{\Delta }d={\mathit{C}}^{-1}\mathrm{\Delta }p$ . By taking the norm of the formula, $||\mathrm{\Delta }d||\le ||{\mathit{C}}^{-1}||\bullet ||\mathrm{\Delta }p||$ can be obtained. Then the following inequality can be solved:

$${\displaystyle \frac{||\mathrm{\Delta }d||}{||{\mathit{C}}^{-1}||\bullet ||p||}\le {\displaystyle \frac{||\mathrm{\Delta }p||}{||p||}}}$$

(22)

According to $p=\mathit{C}d$ , then $||p||\le ||\mathit{C}||\bullet ||d||$ can be obtained. Combining with equation (22), the following inequality can be solved:

$${\displaystyle \frac{1}{||\mathit{C}||\bullet ||{\mathit{C}}^{-1}||}{\displaystyle \frac{||\mathrm{\Delta }d||}{||d||}={\displaystyle \frac{1}{\mathrm{cond}({\mathit{C}}^{-1})}{\displaystyle \frac{||\mathrm{\Delta }d||}{||d||}\le {\displaystyle \frac{||\mathrm{\Delta }p||}{||p||}}}}}}$$

(23)

Finally, combining equations (21) and (23), the following inequalities can be solved:

$${\displaystyle \frac{1}{\mathrm{cond}({\mathit{C}}^{-1})}{\displaystyle \frac{||\mathrm{\Delta }d||}{||d||}\le {\displaystyle \frac{||\mathrm{\Delta }p||}{||p||}\le \mathrm{cond}({\mathit{C}}^{-1}){\displaystyle \frac{||\mathrm{\Delta }d||}{||d||}}}}}$$

(24)

Equation (24) reflects the influences of the disturbance of the control vertex d and the condition number of the control matrix ${\mathit{C}}^{-1}$ on the trajectory point vector. So, when the disturbance of the control vertex d is given, the degree of trajectory deformation can be controlled by the condition number of the control matrix ${\mathit{C}}^{-1}$ . To be specific, the larger the condition number of the control matrix ${\mathit{C}}^{-1}$ is, the larger deformation of trajectory is. Therefore, the following index can be defined to evaluate the trajectory deformation:

$$K_{\mathrm{cond}}=\sqrt{\mathrm{cond}({\mathit{C}}^{-1})}$$

(25)

Based on the above analysis, combined with the jerk index and the trajectory deformation index, the comprehensive evaluation index that considers both the smoothness and the deformation of the trajectory is obtained:

$$K_{\mathrm{com}}=\sqrt{{(K_{\mathrm{jerk}})}^2+{({\omega }_{\mathrm{cond}}{K}_{\mathrm{cond}})}^2}$$

(26)

So the trajectory optimization model for the fifth-order and seventh-order B-spline acceleration laws based on the workspace discretization can be shown as follows:

$$\begin{array}{rl}& \begin{array}{ccc}& & \begin{array}{ccc}& & \begin{array}{ccc}\begin{array}{cc}& \end{array}& \begin{array}{cc}& \end{array}& min\end{array}\end{array}\end{array}{K}_{\mathrm{com}}\\ & \begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& & \end{array}\begin{array}{cc}\mathrm{s}.\mathrm{t}.& \begin{array}{cc}& \end{array}\end{array}{T}_1+T_2=T_{\mathrm{total}}\\ & \begin{array}{ccc}\begin{array}{cc}& \end{array}& & \end{array}{\theta }_{min}-\underset{i\in [1,3]}{min}({\theta }_i)\le 0,\underset{i\in [1,3]}{max}({\theta }_i)-{\theta }_{max}\le 0\\ & \underset{i\in [1,3]}{max}(|v_i|)-v_{Mi,max}\le 0,\underset{i\in [1,3]}{max}(|{\tau }_i|)-{\tau }_{Mi,max}\le 0,\underset{i\in [1,3]}{max}(P_i)-P_{Mi,max}\le 0\end{array}$$

(27)

To analyze the trajectory deformation after comprehensive optimization, taking the trajectory parameters $h_d=0.125\mathrm{m}$ and $\varphi =0^{\circ }$ as an example, the comprehensive optimization is carried out. Considering the smoothness and the deformation of the trajectory comprehensively, the index weight ${\omega }_{\mathrm{cond}}$ is, respectively, set to 300 with the fifth-order B-spline acceleration law and 15 with the seventh-order B-spline acceleration law, and the optimization results are shown in Figure 10. As can be observed in Figure 10, the deformation degree of the trajectory after comprehensive optimization is very small. To be specific, for the fifth-order B-spline and seventh-order B-spline, the maximum deformation values of trajectory are 1.8 mm and 3.8 mm, respectively.

Figure 10.

The motion trajectories after the comprehensive optimization.

After the above comprehensive optimization, for the typical case where $h_d=0.125\mathrm{m}$ and $\varphi =0^{\circ }$ , the final optimal results can be derived as listed in Table 2. Herein, the maximum jerk values of the fifth-order B-spline and seventh-order B-spline is 4.34 × 10⁵°/s³ and 6.78 × 10⁵°/s³, which are lower than that of the typical case. Therefore, for the B-spline acceleration laws, the trajectory planning of the comprehensive index can guarantee both the motion smoothness and the deformation of the trajectory. Furthermore, after optimization, for different acceleration types, the kinematics performances of the robot under the same trajectory parameters are different, so the appropriate acceleration type should be selected according to the actual engineering requirements. In addition, the optimal results obtained through trajectory planning provide a basis for the subsequent simplification analysis of rigid-body dynamics.

Table 2.

The motion parameters of the robot joints before and after comprehensive optimization (units: s, s, °/s, °/s², °/s³, N·m).

	Acceleration type	T 1	T 2	max|v|	max|a|	max|j|	max|τ|
Before optimization	3–4–5 polynomial	0.100	0.150	8.28 × 102	1.92 × 104	1.38 × 106	15.77
	The fifth-order B-spline	0.100	0.150	6.75 × 102	1.35 × 104	5.50 × 105	10.45
	The seventh-order B-spline	0.100	0.150	7.25 × 102	1.53 × 104	7.58 × 105	11.87
After optimization	3–4–5 polynomial	0.073	0.177	7.01 × 102	1.47 × 104	6.82 × 105	11.49
	The fifth-order B-spline	0.079	0.171	6.09 × 102	1.17 × 104	4.34 × 105	8.85
	The seventh-order B-spline	0.089	0.161	6.87 × 102	1.44 × 104	6.78 × 105	10.67

Dynamics model simplification

Simplified rigid-body dynamics model

According to the complete rigid-body dynamics model, the input torque of the joints is composed of four parts, which are acceleration torque term ${\mathit{\tau }}_{\mathrm{com},a}$ , velocity torque term ${\mathit{\tau }}_{\mathrm{com},v}$ , gravity torque term ${\mathit{\tau }}_{\mathrm{com},g}$ , and passive limbs torque term ${\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}$ , respectively. Among them, the passive limb torque term is very complicated, which is not conducive to the online calculation and the design of the dynamics controller. Therefore, it must be simplified properly.

In the complete rigid-body dynamics model, the passive limbs torque term ${\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}$ is expressed as

$${\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}={\mathit{J}}^{-\mathrm{T}}\sum _{i=1}^3({\mathit{J}}_{wi}^{\mathrm{T}}{\mathit{M}}_{\mathrm{rod}i}+{\mathit{J}}_{\mathit{v}i}^{\mathrm{T}}{\mathit{f}}_{si})$$

(28)

where ${\mathit{M}}_{\mathrm{rod}i}=R{I}_{\mathrm{rod}}{R}^T{\omega }_{ai}$ and ${\mathit{f}}_{si}=m_{\mathrm{rod}}{r}_{ca,i}+m_{\mathrm{rod}}g\hat{z}$ . So combined with kinematics analysis, the following equations can be obtained:

$${\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}={\mathit{\tau }}_{\mathrm{rod},a}+{\mathit{\tau }}_{\mathrm{rod},v}+{\mathit{\tau }}_{\mathrm{rod},g}$$

(29)

where

$$\begin{array}{rl}& {\mathit{\tau }}_{\mathrm{rod},a}={\mathit{J}}^{-\mathrm{T}}\sum _{i=1}^3({\mathit{J}}_{wi}^{\mathrm{T}}{I}_{\mathrm{rod}i}{\mathit{J}}_{wi}+{\mathit{J}}_{\mathit{v}i}^{\mathrm{T}}{m}_{\mathrm{rod}}{\mathit{J}}_{\mathit{v}i})r_a\\ & {\mathit{\tau }}_{\mathrm{rod},v}={\mathit{J}}^{-\mathrm{T}}\sum _{i=1}^3({\mathit{J}}_{wi}^{\mathrm{T}}{I}_{\mathrm{rod}i}{K}_i(r_v)+{\mathit{J}}_{\mathit{v}i}^{\mathrm{T}}{m}_{\mathrm{rod}}{N}_i(r_v))\\ & {\mathit{\tau }}_{\mathrm{rod},g}={\mathit{J}}^{-\mathrm{T}}\sum _{i=1}^3({\mathit{J}}_{\mathit{v}i}^{\mathrm{T}}{m}_{\mathrm{rod}}g\hat{z})\end{array}$$

According to equation (29), the passive torque term ${\mathit{\tau }}_{\mathrm{com},\mathrm{rod}}$ is also composed of the acceleration torque term ${\mathit{\tau }}_{\mathrm{rod},a}$ , velocity torque term ${\mathit{\tau }}_{\mathrm{rod},v}$ , and gravity torque term ${\mathit{\tau }}_{\mathrm{rod},g}$ , which are affected by the movement and configuration of the robot.

Generally, without considering the different torque components, the idea of rigid-body dynamics model simplification is directly divided the mass of the passive limbs into two parts, which are attached to the active limbs and the moving platform, respectively. And the mass distribution coefficients are r and 1 − r, respectively, which are usually unchanged for different trajectory parameters and acceleration types. But for different acceleration types, the contribution ratios of different torque components of the passive limb are different, as shown in Figure 11. Therefore, without considering the different torque components, taking the single fixed mass distribution coefficient for different trajectory parameters and acceleration types, the simplified model will produce a large deviation. A combined mass distribution coefficients simplification method, which considers the different torque components, is proposed in this section.

Figure 11.

The variation of torque components of the passive limb in a single period.

Therefore, according to the acceleration term, velocity term, and gravity term, the mass of the passive limb is independently equivalent to the active limb and the moving platform, respectively. So the simplified rigid-body dynamics model based on the combined mass distribution coefficients can be obtained

$${\mathit{\tau }}_{\mathrm{sim}}={\mathit{\tau }}_{\mathrm{sim},a}+{\mathit{\tau }}_{\mathrm{sim},v}+{\mathit{\tau }}_{\mathrm{sim},g}$$

(30)

where

$$\begin{array}{rl}& {\mathit{\tau }}_{\mathrm{sim},a}=(m_a{\mathit{J}}^{-\mathrm{T}}+I_{A,a}\mathit{J})r_a\\ & {\mathit{\tau }}_{\mathrm{sim},v}=I_{A,v}\mathit{f}(r_v)\\ & {\mathit{\tau }}_{\mathrm{sim},\mathrm{g}}=m_{A,g}{r}_{A,g}g{\left(\begin{array}{ccc}\cos {\theta }_1& \cos {\theta }_2& \cos {\theta }_3\end{array}\right)}^{\mathrm{T}}+m_{g}g{\mathit{J}}^{-\mathrm{T}}\hat{z}\end{array}$$

And the equivalent inertia parameters of the simplified rigid-body dynamics model based on the combined mass distribution coefficients are shown in Table 3.

Table 3.

The equivalent inertia parameters of the simplified rigid-body dynamics model.

Symbol	Meaning
$I_{A,a}=I_{\mathrm{arm}}+r_{a1}{m}_{\mathrm{rod}}{l}_1^2+n^2{I}_{\mathrm{gear}}$	The equivalent rotational inertia of acceleration term of active arms
$I_{A,v}=I_{\mathrm{arm}}+r_v{m}_{\mathrm{rod}}{l}_1^2+n^2{I}_{\mathrm{gear}}$	The equivalent rotational inertia of velocity term of active arms
$m_a=m_{\mathrm{platform}}+3r_{a2}{m}_{\mathrm{rod}}+m_{\mathrm{load}}$	The equivalent mass of acceleration term of the moving platform
$m_g=m_{\mathrm{platform}}+3r_{g2}{m}_{\mathrm{rod}}+m_{\mathrm{load}}$	The equivalent mass of gravity term of the moving platform
$m_{A,g}{r}_{A,g}=m_A{r}_A+r_{g1}{m}_{\mathrm{rod}}{l}_1$	The equivalent mass–radius product of gravity term of active arms

Coefficients determination

To determine the value of the five mass distribution coefficients ( $r_{a1}$ , $r_{a2}$ , $r_v$ , $r_{g1}$ , and $r_{g2}$ ) reasonably, an optimization model for determining the mass distribution coefficients is established in this section. By analyzing the error between the complete model and the simplified model, two indexes for evaluating the accuracy of the simplified model are proposed

$$\begin{array}{rl}& \mathrm{\Delta }{\tau }_{\mathrm{mean}}={\displaystyle \frac{1}{3}\sum _{i=1}^3{\displaystyle \frac{{\int }_0^{T_{\mathrm{total}}}|{\tau }_{\mathrm{com},i}(t)-{\tau }_{\mathrm{sim},i}(t)|\mathrm{d}t}{{\int }_0^{T_{\mathrm{total}}}\mathrm{d}t}}}\\ & \mathrm{\Delta }{\tau }_{max}=\underset{i\in [1,3]}{max}\underset{t\in [0,T_{\mathrm{total}}]}{max}(|{\tau }_{\mathrm{com},i}(t)-{\tau }_{\mathrm{sim},i}(t)|)\end{array}$$

(31)

where $\mathrm{\Delta }{\tau }_{\mathrm{mean}}$ is the mean value of the torque error between the complete dynamics model and the simplified dynamics model in a single motion period, $\mathrm{\Delta }{\tau }_{max}$ is the max value of the torque error among three joints in a single motion period, ${\tau }_{\mathrm{com},i}(t)$ and ${\tau }_{\mathrm{sim},i}(t)$ represent the ith joint torque values calculated by the complete rigid-body dynamics model and the simplified rigid-body dynamics model, respectively.

Based on the accuracy evaluation indexes, the optimization model for dynamics model simplification under the given trajectory parameters and acceleration type can be expressed as follows:

$$\begin{array}{rl}& \begin{array}{ccc}& & \end{array}min\mathrm{\Delta }{\tau }_f={\omega }_1\mathrm{\Delta }{\tau }_{\mathrm{mean}}+{\omega }_2\mathrm{\Delta }{\tau }_{max}\\ & \mathrm{s}.\mathrm{t}.{\omega }_1+{\omega }_2=1,0\le {\omega }_1,{\omega }_2,r_{a1},r_{a2},r_v,r_{g1,}{r}_{g2}\le 1\end{array}$$

(32)

where ${\omega }_1$ and ${\omega }_2$ are the weight coefficients of $\mathrm{\Delta }{\tau }_{\mathrm{mean}}$ and $\mathrm{\Delta }{\tau }_{max}$ , the value can be adjusted according to the simplification demand. For example, increasing ${\omega }_1$ tends to optimize the mean value of torque error $\mathrm{\Delta }{\tau }_{\mathrm{mean}}$ , on the contrary, increasing ${\omega }_2$ tends to optimize the max value of torque error $\mathrm{\Delta }{\tau }_{max}$ . Based on the synthesis consideration of $\mathrm{\Delta }{\tau }_{\mathrm{mean}}$ and $\mathrm{\Delta }{\tau }_{max}$ , taking ${\omega }_1={\omega }_2=0.5$ as an example in this section.

To make the determination process of combined mass distribution coefficients clear, a flowchart is depicted in Figure. 12. Firstly, by the trajectory planning based on the workspace discretization, the optimal segment time under different trajectory parameters is obtained. Secondly, the simplified rigid-body dynamics model based on combination coefficients and a complete rigid-body dynamics model are established. Finally, based on the above researches, analyzing the difference between complete and simplified dynamics models whether can meet the mission requirements. When the TC is satisfied, the final optimal combination coefficients can be obtained.

Figure 12.

The determination procedure of combined mass distribution coefficients.

According to the trajectory planning results, the kinematics performances are greatly inflected by h_d. Therefore, to analyze the influence of the trajectory parameters on the combined mass distribution coefficients, when the segment time of trajectory optimization under different acceleration types is adopted, the relationship between the combined mass distribution coefficients and $h_d$ (i.e. taking $\varphi =0^o$ as an example) is depicted in Figure 13. As can be observed, the mass distribution coefficients can be mainly affected by the trajectory parameters. To be specific, for different acceleration types, $r_{a1}$ shows an upward trend with $h_d$ , but $r_{a2}$ shows a downward trend with $h_d$ . Meanwhile, r_v remains almost unchanged with the increase of h_d. But the r_g1 and r_g2 show a fluctuating trend with the increase of h_d. According to the previous kinematics analysis, the angular motion range of the active limbs will increase with the increase of h_d in a single movement period. Therefore, the acceleration of the active limb will increase accordingly. So the proportion of the acceleration force term caused by the motion of the active limbs will increase. In addition, for the same trajectory parameters, when different acceleration types are adopted, the mass distribution coefficients are different.

Figure 13.

The relationship between the r and h_d.

Taking the final optimal segmentation time of trajectory optimization in Table 2 as an example, the optimization results of the combined mass distribution coefficients are shown in Table 4. To evaluate the accuracy of the simplified dynamics model based on the combined coefficients, a comparative analysis of the simplified dynamics model based on the single coefficient, where the mass distribution coefficient r is 2/3 or 0.7, respectively, is carried out in this section. As can be observed in Figure 14, for the different acceleration types, the accuracy of the simplified rigid-body dynamics model based on the combined coefficients is significantly higher than that based on the single coefficient. To be specific, for 3–4–5 polynomial, the fifth-order B-spline and the seventh-order B-spline, $\mathrm{\Delta }{\tau }_{\mathrm{mean}}$ of the combined mass distribution coefficients are 0.1696 N·m, 0.1509 N·m, and 0.1705 N·m, respectively, and $\mathrm{\Delta }{\tau }_{max}$ of the combined mass distribution coefficients are 0.3238 N·m, 0.2698 N·m, and 0.3089 N·m, respectively. In brief, compared with the simplified dynamics model based on the single coefficient, $\mathrm{\Delta }{\tau }_{\mathrm{mean}}$ and $\mathrm{\Delta }{\tau }_{max}$ of the simplified dynamics model based on the combined coefficients model are significantly reduced.

Figure 14.

Comparison of different simplified methods.

Table 4.

The simplification results of the combined mass distribution coefficients.

Acceleration type	ra 1	ra 2	rv	rg 1	rg 2	Δτmean (N·m)	Δτmax (N·m)
3–4–5 polynomial	0.6842	0.2966	0.3945	0.2511	0.6909	0.1696	0.3238
The fifth-order B-spline	0.6696	0.2973	0.3842	0.2367	0.6990	0.1509	0.2698
The seventh-order B-spline	0.6748	0.2931	0.3971	0.2427	0.6894	0.1705	0.3089

Conclusions

This paper proposed a hierarchical approach for rigid-body dynamics model simplification of the high-speed parallel robot by considering kinematics performance. The main conclusions are drawn as follows:

Considering the effects of the trajectory parameters and the acceleration types on the robot output performance, a new trajectory planning method based on the workspace discretization was proposed. And for the minimum jerk optimization, the influence of trajectory parameters on segmentation time is different, but the influence of trajectory parameters on the smoothness index is almost the same.

For the fifth-order and seventh-order B-spline acceleration types, by taking trajectory smoothness index and trajectory deformation index into account, a comprehensive optimization method was proposed. And this method can reduce the maximum jerk value of the robot joint under the allowable deformation of the trajectory curve.

By decomposing the torque of the passive limbs, a dynamics model simplification method based on the combined mass coefficients was proposed. And comparing with the simplification method based on the single mass distribution coefficient, the simplification method based on the combined coefficients can greatly reduce the simplification error under different trajectory parameters and acceleration types.

Therefore, the hierarchical approach can guarantee both the excellent kinematics performance of the robot and the accuracy of the simplified dynamics model under different trajectory parameters and acceleration types. And the simplified model can be implemented in the design of the dynamics controller to enhance the robot performance. However, considering the calculation rapidity, the combined mass distribution coefficients are not adjusted in real-time according to the motion of the robot, which will be analyzed in future research work.