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Abstract

An approach to the dynamic modeling and sliding mode control of the constrained robot is proposed in this article. On the basis of the Udwadia–Kalaba approach, the explicit equation of the constrained robot system is obtained first. This equation is applicable to systems with either holonomic or non-holonomic constraints, as well as with either ideal or non-ideal constraint forces. Second, fully considering the uncertainty of the non-ideal force, that is, the dynamic friction in the constrained robot system, the sliding mode control algorithm is put forward to trajectory tracking of the end-effector on a vertical constrained surface to obtain actual values of the unknown constraint force. Moreover, model order reduction method is innovatively used in the Udwadia–Kalaba approach and sliding mode controller to reduce variables and simplify the complexity of the calculation. Based on the demonstration of this novel method, a detailed robot system example is finally presented.

Keywords

Constrained robot Udwadia–Kalaba equation sliding mode control dynamic modeling simulation

Introduction

A constrained robot system is a typical mechanical system. The control of this kind of system usually needs some dynamic equations. It is known that the robot system has characteristics of high coupling, nonlinearity, and uncertainty in trajectory tracking. As a result, it is almost impossible to build a model of the robot dynamics perfectly. Fortunately, this solution of this problem is vigorously worked since the constrained movement technique was proposed by Lagrange.¹ He developed a Lagrange multiplier method for solving constrained movements. However, in practical engineering applications, it is difficult to get the Lagrange multiplier, which leads to the difficulty of obtaining this equation. Then, Gauss² provided a new common principle for motions of constrained mechanical systems, which can be applied in constrained robot systems. Gibbs³ and Appell⁴ also present the comprehensive equation of movement which is apprised highly by Pars;⁵ however, the equation is difficult to deal with large degree of freedom (DOF).

Professors Udwadia and Kalaba^6–9 proposed the equation of the multi-body system motion under the constraint condition, which is one of the important achievements in Lagrange mechanics field. This equation is applicable to a variety of constraints, such as holonomic and non-holonomic constraints. Later on, they extended their work to the non-ideal constraint system and general mechanical system. The merit of their method is that the system they focus on may not meet D’Alembert’s principle,¹⁰ while other works are almost on the basis of D’Alembert’s principle. Attributing to the simple and general expression, this equation has attracted more and more attentions and has been applied in many different fields.

In recent years, researchers have done much work to obtain the dynamic model and control of the constrained robot. Liu and Liu¹¹ got dynamic modeling of industrial robot subject to constraint using the Udwadia–Kalaba equation, and the ideal constraint force is taken into consideration. Su et al.¹² used a sliding mode control algorithm in the constrained robot system, but the approach ignored the constraint force which is indispensable in practical application. Wang et al.¹³ primarily studied the variable control of non-holonomic constraints. Wanichanon et al.¹⁴ proposed a general sliding control scheme on the holonomic and non-holonomic constraints. On the foundation of their work, the ideal force and non-ideal force are both taken into consideration and a novel dynamic model is established.

Actually, a constrained robot system usually suffers from the constraint force, which is commonly caused by the end-effector of the robot being constrained on a surface. In practice, the constraint force which is produced on the constraint surface cannot be ignored. This constraint force can be divided into two parts. One is the normal force which is regarded as the ideal constraint force. Thanks to the Udwadia–Kalaba approach, it can be used to obtain the dynamic model combined with normal force in constrained robot system, and the explicit expression of the normal force can be obtained. The other is the tangential force which can be seen as the non-ideal constraint force. Non-ideal constraint forces often include friction force and electromagnetic force. Note that the friction force cannot be ignored in the constrained robot system, but unfortunately it cannot be calculated. In the simulation, the friction force can be obtained by first giving an initial condition and then introducing the sliding mode control method to track the trajectory and force.

The constrained robot system is a very complicated multi-input multi-output (MIMO) nonlinear system, which has several dynamic characteristics of time-varying, coupling, and nonlinear. For these characteristics, neural adaptive control and sliding mode control are regularly used to control the constrained robot, and lots of researchers have achieved good results. S Frikha et al.¹⁵ proposed an adaptive neural sliding mode control scheme with Lyapunov criterion for typical uncertain nonlinear systems, and neural network was used to estimate the structural model of the system. H Wei et al.¹⁶ used adaptive neural network control with full-state feedback for an uncertain constrained robot, which can effectively guarantee the performance and improve the robustness of closed-loop system. R García-Rodríguez and V Parra-Vega¹⁷ designed a neural sliding mode control scheme for constrained robots based on Lyapunov function, which can prove the robustness of closed-loop system and finally conclude convergence of position and force tracking errors. Liu et al.¹⁸ proposed neural network control which is based on adaptive learning design for nonlinear systems with state constraints, in which signals of the closed-loop system are bounded and the tracking error converges to a bounded compact set.

In this article, the sliding mode control^19–21 is used to trajectory tracking of the end-effector on a vertical constrained surface to obtain actual values of the unknown constraint force. The sliding mode control is also called the variable structure control, which is proposed by Soviet scholars Utkin and Emeleyanov. The structure of the sliding mode control system is constantly changing as the current state, so that the system is moving according to a predetermined trajectory. The sliding mode control method is suitable for the robot control because of its two benefits. On one hand, the sliding mode control does not need the accurate mathematical model of the controlled object. As mentioned above, in the constrained robot system, the non-ideal force can only be obtained by experimentation or observation. Therefore, utilizing this benefit, the sliding mode control algorithm is appropriate to achieve trajectory and force tracking. On the other hand, the sliding mode control is invariant to uncertainty factors such as parameters perturbation and the external disturbance. This benefit can ensure the control performance of the system due to the random interference. The main contributions of this article are as follows:

By the Udwadia–Kalaba equation, the ideal and non-ideal forces are both taken into consideration. Because the non-ideal constraint force is hardly calculated but only can be obtained by the experiment or the experience, the sliding mode control algorithm is developed for tracking the non-ideal force (the dynamic friction) in the constrained robot system. In this way, the dynamic equation of the constrained robot is more complete and the non-ideal force can be obtained theoretically. This contribution gives a theoretical basis for the future experiment investigation.

The major innovation of this article is the establishment of a new order reductive dynamic equation and sliding mode controller to describe the constrained robot motion. The model order reduction method not only can simplify the complexity of the calculation but also can be extended to multi-degree of the constrained robot system.

The outline of this article is organized as follows. First, Udwadia–Kalaba approach is described in detail. Second, the dynamic model of the constrained robot system is obtained by the order reduction and Udwadia–Kalaba approach and the sliding mode control algorithm is derived. Third, the 2-DOF robot with vertical constraint is used as the example to specify and verify the correctness of the Udwadia–Kalaba approach and sliding mode control approach. Fourth, some conclusions are presented.

Detail the Udwadia–Kalaba approach

To the robot system without constraint, the dynamic equation of n-DOF robot is established with Lagrange method²²

M (q, t) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) = τ

(1)

where $q = [q_{1}, q_{2}, \dots, q_{n}]^{T}$ describes joint displacements. $τ$ denotes applied joint torques. $M (q, t)$ is the symmetric and positive inertia or mass matrix. $C (q, \overset{\cdot}{q})$ represents coriolis and centrifugal torques. $G (q)$ denotes gravitational torques. Here, assume joint displacements are independent of each other. Equation (1) is written in the following form

M (q, t) \overset{\cdot\cdot}{q} = Q (q, \overset{\cdot}{q}, t)

(2)

$Q (q, \overset{\cdot}{q}, t)$ can be thought of the external resultant force of the constraint system. From equation (2), when $q, \overset{\cdot}{q}, and t$ are known, the acceleration can be obtained as follows

a (q, \overset{\cdot}{q}, t) = M^{- 1} (q, t) Q (q, \overset{\cdot}{q}, t)

(3)

It is assumed that the constraint form of this robot system can be described by $m = m_{1} + m_{2}$ equations⁶

φ_{i} (q, t) = 0 i = 1, 2, \dots, m_{1}

(4)

and

ψ_{j} (q, \overset{\cdot}{q}, t) = 0 j = 1, 2, \dots, m_{2}

(5)

where $φ$ is a $m_{1}$ vector and $ψ$ is a $m_{2}$ vector. Equations (4) and (5) include all the usual varieties of holonomic and/or non-holonomic constraints. Differentiating equation (4) twice with respect to time and equation (5) once with respect to time, the following matrix form²³ can be obtained

A (q, \overset{\cdot}{q}, t) \overset{\cdot\cdot}{q} = b (q, \overset{\cdot}{q}, t)

(6)

where A referred to as $m \times n$ constraint matrix and b is a m-vector.

When the system is constrained and additional set of forces act on the robot system, which can be called the constrained system, the motion equation of the constrained robot system is given by

M (q, t) \overset{\cdot\cdot}{q} = Q (q, \overset{\cdot}{q}, t) + Q^{c} (q, \overset{\cdot}{q}, t)

(7)

where $Q^{c} (q, \overset{\cdot}{q}, t)$ is n-vector, which is caused by the additional constraint force and satisfies constraint conditions.

According to D’Alembert’s principle, constraint forces can do positive, negative, or zero work under virtual displacement in the constrained system. When constraint forces do no work, they are called ideal constraint forces. While constraint forces do work, they can be named as the non-ideal constraint force which is the dynamic friction in this article. Therefore, when the constrained robot system exists ideal and non-ideal constraints in the same time, the $Q^{c} (q, \overset{\cdot}{q}, t)$ can be given by

Q^{c} (q, \overset{\cdot}{q}, t) = Q_{id}^{c} (q, \overset{\cdot}{q}, t) + Q_{nid}^{c} (q, \overset{\cdot}{q}, t)

(8)

where $Q_{id}^{c} (q, \overset{\cdot}{q}, t)$ denotes the ideal constraint force and $Q_{nid}^{c} (q, \overset{\cdot}{q}, t)$ represents the non-ideal constraint force.

Assuming that the virtual displacement²⁴ is $υ$ , the work done by the ideal constraint force $Q_{id}^{c}$ is shown as

υ^{T} Q_{id}^{c} = 0

(9)

While the work done by the non-ideal constraint force $Q_{nid}^{c}$ is

υ^{T} Q_{nid}^{c} \neq 0

(10)

The form of the ideal and non-ideal constraint force has been given by Udwadia and Kalaba⁶

Q_{id}^{c} = M^{\frac{1}{2}} B^{+} (b - A M^{- 1} Q)

(11)

and

Q_{nid}^{c} = M^{\frac{1}{2}} (I - B^{+} B) M^{- \frac{1}{2}} c

(12)

where $B = A M^{- 1 / 2}$ and the superscript “+” represents the Moore–Penrose inverse matrix. The vector c is a known vector, which can be obtained by experimentation or observation in a certain mechanical system.⁸

From above all, the equation of the constrained robot system is given by

M \overset{\cdot\cdot}{q} = Q + M^{\frac{1}{2}} B^{+} (b - A M^{- 1} Q) + M^{\frac{1}{2}} (I - B^{+} B) M^{- \frac{1}{2}} c

(13)

Remark

Non-holonomic constraint is the constraint that contains time derivatives of the generalized coordinates of the system, which is not integrable. While, non-ideal constraint is the one that does virtual work which is not equal to zero in any particle system. So, non-holonomic and non-ideal constraints are naturally different in the aspect of definition. In this article, holonomic and non-holonomic constraints, as well as ideal and non-ideal constraints are used to indicate different kinds of constraints in the constrained robot systems. And according to the Udwadia–Kalaba approach, the explicit equation of the constrained robot system is applicable to all holonomic and non-holonomic (ideal and non-ideal) constrained systems no matter whether they satisfy D’Alembert’s principle.

Model reduction

The constrained robot is a typical mechanical system. The 2-DOF robot with vertical constraints is shown in Figure 1 below.

Figure 1.

2-DOF robot with vertical constraint.

As shown in Figure 1, it is the schematic diagram of 2-DOF robot with the vertical constraint. Let $x = [x_{1}, x_{2}]^{T}$ denotes the end-effector coordinate of the constrained robot in Cartesian coordinate system. $q = [q_{1}, q_{2}]^{T}$ is the generalized coordinate of the system. There are two perpendicular forces acting on the end-effector by the vertical plane. The first force is the ideal constraint force $Q_{id}^{c}$ , which is the positive pressure on the contact surface and is perpendicular to the constrained surface. The second force is the non-ideal constraint force $Q_{nid}^{c}$ , which remains tangent to the constrained surface. Therefore, the ideal constraint force $Q_{id}^{c}$ provides holding power to guarantee the end-effector for moving on the contact surface. The non-ideal constraint force $Q_{nid}^{c}$ provides the tangential acceleration along the constrained surface.

The equation of the constraint is written as²⁰

ϕ (x) = 0

(14)

where $ϕ$ is two times continuously differentiable. Assuming that the vector x and q have such a relation

x = h (q)

(15)

where h is two times continuously differentiable, then the equation of constraints in joint space is obtained

Φ (q) = ϕ (h (q)) = 0

(16)

In the constrained robot system, $Q_{id}^{c}$ can be obtained by equation (11). $Q_{nid}^{c}$ includes the dynamic friction only in the end-effector. When the end-effector is moving on the vertical plane, it is the dynamic friction acting on the robot. Due to the uncertainty of the $Q_{nid}^{c}$ , the non-ideal constraint force is expressed as

Q_{nid}^{c} = J^{T} (q, t) F (t) = J^{T} (q, t) λ

(17)

where $F (t)$ is the dynamic friction in Cartesian space. $λ$ is the associated Lagrangian multiplier. $J^{T} (q, t)$ is the Jacobian matrix of equation (17), which can be given by

J (q, t) = \frac{\partial Φ (q)}{\partial q} = \frac{\partial ϕ (x)}{\partial x} \frac{\partial x}{\partial q} = \frac{\partial ϕ (x)}{\partial x} \frac{\partial h (q)}{\partial q}

(18)

From equations (1), (7), (8), and (17), the dynamic equation of the 2-DOF constrained robot is given by

M (q, t) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + Q_{id}^{c} (q, \overset{\cdot}{q}, t) + J^{T} (q, t) λ = τ

(19)

Since the end-effector of the robot is constrained in the vertical surface, DOFs of the robot system changed from two to one. Here, choose $q_{1}$ as the variable to describe the motion of the constrained robot. So, $q_{2}$ is the remaining joint redundant variable. And $q_{2}$ can be donated by $q_{1}$

q_{2} = ψ (q_{1})

(20)

And then from equation (20), one can obtain

\overset{\cdot}{q} = [\begin{matrix} {\overset{\cdot}{q}}_{1} \\ {\overset{\cdot}{q}}_{2} \end{matrix}] = [\begin{matrix} {\overset{\cdot}{q}}_{1} \\ \frac{\partial ψ (q_{1})}{\partial q_{1}} {\overset{\cdot}{q}}_{1} \end{matrix}] = L (q_{1}) {\overset{\cdot}{q}}_{1}

(21)

and

\overset{\cdot\cdot}{q} = \dot{L} (q_{1}) {\dot{q}}_{1} + L (q_{1}) {\overset{\cdot\cdot}{q}}_{1}

(22)

where $L (q_{1}) = [\begin{matrix} 1 \\ \partial ψ (q_{1}) / \partial q_{1} \end{matrix}]$ .

Therefore, equation (19) is expressed in the reduced form as

M_{1} (q_{1}, t) {\overset{\cdot\cdot}{q}}_{1} + C_{1} (q_{1}, {\overset{\cdot}{q}}_{1}) {\overset{\cdot}{q}}_{1} + G_{1} (q_{1}) + Q_{{id}_{1}}^{c} (q_{1}, {\overset{\cdot}{q}}_{1}, t) + J^{T} (q_{1}, t) λ = τ

(23)

where $M_{1} (q_{1}, t) = M (q, t) L (q_{1})$ , $C_{1} (q_{1}, {\overset{\cdot}{q}}_{1}) = M (q, t) \overset{\cdot}{L} (q_{1}) + C (q, \overset{\cdot}{q}) L (q_{1})$ , $G_{1} (q_{1}) = G (q)$ , $Q_{id 1}^{c} (q_{1}, {\overset{\cdot}{q}}_{1}, t) = Q_{id}^{c} (q, \overset{\cdot}{q}, t)$ , and $J^{T} (q_{1}, t) = J^{T} (q, t)$ .

Remark

Equation (23) is the basis for the control purpose of the constrained robot system.

Now multiply both sides of equation (23) with $L^{T} (q_{1})$ , the following equation is obtained

\begin{matrix} L^{T} (q_{1}) M_{1} (q_{1}, t) {\overset{\cdot\cdot}{q}}_{1} + L^{T} (q_{1}) C_{1} (q_{1}, {\overset{\cdot}{q}}_{1}) {\overset{\cdot}{q}}_{1} + L^{T} (q_{1}) G_{1} (q_{1}) \\ + L^{T} (q_{1}) Q_{{id}_{1}}^{c} (q_{1}, {\overset{\cdot}{q}}_{1}, t) \\ + L^{T} (q_{1}) J^{T} (q_{1}, t) λ = L^{T} (q_{1}) τ \end{matrix}

(24)

Equation (24) can be simplified as

M_{L} (q_{1}, t) {\overset{\cdot\cdot}{q}}_{1} + C_{L} (q_{1}, {\dot{q}}_{1}) {\dot{q}}_{1} + G_{L} (q_{1}) + Q_{i d_{L}}^{c} (q_{1}, {\dot{q}}_{1}, t) = L^{T} τ

(25)

By exploiting the structure of equations (23) and (25), three properties can be obtained:²⁵

Property 1: Define the matrix $M_{L} (q_{1}, t) = L^{T} (q_{1}) M_{1} (q_{1}, t), M_{L} (q_{1}, t) > 0$ .

Property 2: Define the matrix $C_{L} (q_{1}, {\overset{\cdot}{q}}_{1}) = L^{T} (q_{1}) C_{1} (q_{1}, {\overset{\cdot}{q}}_{1})$ and then ${\overset{\cdot}{M}}_{L} (q_{1}, t) - 2 C_{L} (q_{1}, {\overset{\cdot}{q}}_{1})$ is the skew symmetric matrix.

Property 3: $J (q_{1}, t) L (q_{1}) = L^{T} (q_{1}) J^{T} (q_{1}, t) = 0$ .

Three properties are the basis of the design for the sliding mode control laws.

To obtain the practical dynamic friction in simulation, the $λ$ value can be obtained using the following three situations.

Situation 1: When $J (q) = 0$ , the value of $λ$ can be obtained by the expression of $J^{T} (q_{1}, t)$ or the definition of the dynamic friction.

Situation 2: When $J (1) \neq 0$ , the value of $λ$ is given by

λ = \frac{1}{J (1)} (τ (1) - M_{1} (1) {\overset{\cdot\cdot}{q}}_{1} - C_{1} (1) {\overset{\cdot}{q}}_{1} - G_{1} (1) - Q_{id 1}^{c} (1))

(26)

Situation 3: When $J (2) \neq 0$ , the value of $λ$ is given by

λ = \frac{1}{J (2)} (τ (2) - M_{1} (2) {\overset{\cdot\cdot}{q}}_{1} - C_{1} (2) {\overset{\cdot}{q}}_{1} - G_{1} (2) - Q_{id 1}^{c} (2))

(27)

To be sure, the value of $λ$ can be measured by the force sensor in the practical engineering.

Sliding mode control for the constrained robot system

In this section, a general tracking problem for the constrained robot system is considered. As the desired joint position $q_{d} (t)$ and the desired constraint force $λ_{d}$ are known, the desired constraint force which is known for the desired dynamic friction $J^{T} (q, t) λ_{d}$ is known. Note that the objective of the control is to track the desired joint position and the constraint force within an acceptable error range and to satisfy the imposed constraints, $Q_{nidd}^{c} = J^{T} (q, t) λ_{d}$ and $Φ (q_{d}) = 0$ . A sliding mode control law is designed to make $q (t)$ track $q_{d} (t)$ , and $Q_{nid}^{c}$ track $Q_{nidd}^{c}$ as $t \to \infty$ . Since $q_{2} = ψ (q_{1})$ , a sliding mode control law is only need to be found to satisfy $q_{1} (t)$ track $q_{d 1} (t)$ as $t \to \infty$ .

Defining

e_{1} = q_{d 1} - q_{1}

(28)

{\overset{\cdot}{q}}_{r 1} = {\overset{\cdot}{q}}_{d 1} + Λ e_{1}

(29)

e_{λ} = λ_{d} - λ

(30)

where $e_{1}$ denotes the tracking error, $e_{λ}$ represents the force tracking error, $q_{r 1}$ is the reference trajectory, and $Λ > 0$ is the tunable matrix.

The sliding surface is defined as

s_{1} = {\overset{\cdot}{q}}_{r 1} - {\overset{\cdot}{q}}_{1} = {\overset{\cdot}{e}}_{1} + Λ e_{1}

(31)

s_{L 1} = L (q_{1}) s_{1}

(32)

The sliding controller is defined as

τ = M_{1} (q_{1}, t) {\overset{\cdot\cdot}{q}}_{r 1} + C_{1} (q_{1}, {\overset{\cdot}{q}}_{1}) {\overset{\cdot}{q}}_{r 1} + G_{1} (q_{1}) + Q_{{id}_{1}}^{c} (q_{1}, {\overset{\cdot}{q}}_{1}, t) + K_{p} s_{L 1} + J^{T} (q_{1}, t) λ_{r}

(33)

where $K_{p} > 0$ .

The item for controlling the dynamic friction is given by

λ_{r} = λ_{d} + K_{λ} e_{λ}

(34)

where $K_{λ} > 0$ .

From equation (34), the following equation is obtained

λ_{r} - λ = e_{λ} + K_{λ} e_{λ} = (1 + K_{λ}) e_{λ}

(35)

Using equations (23), (31), and (33), the following equation can be obtained

M_{1} (q_{1}, t) {\overset{\cdot}{s}}_{1} + C_{1} (q_{1}, {\overset{\cdot}{q}}_{1}) s_{1} + K_{p} s_{L 1} = J^{T} (q_{1}, t) (λ - λ_{r})

(36)

Multiply both sides of equation (36) with $L^{T} (q_{1})$ and using the Property 3, equation (36) can be written as

M_{L} (q_{1}, t) {\overset{\cdot}{s}}_{1} + C_{L} (q_{1}, {\overset{\cdot}{q}}_{1}) s_{1} + L^{T} (q_{1}) K_{p} s_{L 1} = 0

(37)

The Lyapunov function is taken as

V = \frac{1}{2} M_{L} (q_{1}, t) s_{1}^{2}

(38)

Differentiating equation (38) with respect to time gives

\overset{\cdot}{V} = s_{1} (M_{L} (q_{1}, t) {\overset{\cdot}{s}}_{1} + \frac{1}{2} {\overset{\cdot}{M}}_{L} (q_{1}, t) s_{1})

(39)

Considering the skew symmetric property of ${\overset{\cdot}{M}}_{L} (q_{1}, t) - 2 C_{L} (q_{1}, {\overset{\cdot}{q}}_{1})$ , equation (39) is expressed as

\overset{\cdot}{V} = s_{1} (M_{L} (q_{1}, t) {\overset{\cdot}{s}}_{1} + C_{L} (q_{1}, {\overset{\cdot}{q}}_{1}) s_{1})

(40)

Substituting equation (37) into equation (40), one can obtain

\begin{matrix} \overset{\cdot}{V} = s_{1} (M_{L} (q_{1}, t) {\overset{\cdot}{s}}_{1} + C_{L} (q_{1}, {\overset{\cdot}{q}}_{1}) s_{1}) = s_{1} (- L^{T} (q_{1}) K_{p} s_{L 1}) \\ = - s_{L 1}^{T} K_{p} s_{L 1} = - K_{p} s_{L 1}^{2} \leq 0 \end{matrix}

(41)

Since $\overset{\cdot}{V}$ is negative semi-definite and $K_{p}$ is positive definite, when $\overset{\cdot}{V} \equiv 0$ , the following are obtained

\begin{matrix} s_{L 1} \equiv 0, {\overset{\cdot}{s}}_{L 1} \equiv 0 \\ s_{1} \equiv 0, {\overset{\cdot}{s}}_{1} \equiv 0 \\ {\overset{\cdot}{e}}_{1} \equiv e_{1} \equiv 0 \end{matrix}

(42)

According to equations (35), (36), and (42), it is obvious that

\begin{matrix} λ - λ_{r} \equiv 0 \\ e_{λ} \equiv 0 \end{matrix}

(43)

The following can be known by LaSalle theorem

e_{1} \to 0, {\overset{\cdot}{e}}_{1} \to 0, λ \to λ_{d} as t \to \infty

Simulated example

As shown in Figure 1, the 2-DOF robot with vertical constraint is used to verify the correctness and reliability of the proposed dynamic model and the sliding mode control method. Detailed matrices in equation (1) are shown in the following

M (q, t) = [\begin{matrix} p_{1} + p_{2} + p_{3} + 2 p_{4} \cos q_{2} & p_{3} + p_{4} \cos q_{2} \\ p_{3} + p_{4} \cos q_{2} & p_{3} \end{matrix}]

(44)

C (q, \overset{\cdot}{q}) = [\begin{matrix} - p_{4} {\overset{\cdot}{q}}_{2} \sin q_{2} & - p_{4} ({\overset{\cdot}{q}}_{1} + {\overset{\cdot}{q}}_{2}) \sin q_{2} \\ p_{4} {\overset{\cdot}{q}}_{1} \sin q_{2} & 0 \end{matrix}]

(45)

G (q) = [\begin{matrix} \frac{3}{2} p_{1} e_{1} \cos q_{1} + p_{2} e_{1} \cos q_{1} + p_{4} e_{1} \cos (q_{1} + q_{2}) \\ p_{4} e_{1} \cos (q_{1} + q_{2}) \end{matrix}]

(46)

where $e_{1} = g / l_{1}$ , g is the acceleration of gravity. And the four unknown parameters $p_{1}$ , $p_{2}$ , $p_{3}$ , and $p_{4}$ are functions of physical parameters

\begin{matrix} p_{1} = \frac{1}{3} m_{1} l_{1}^{2} \\ p_{2} = m_{2} l_{1}^{2} \\ p_{3} = \frac{1}{3} m_{2} l_{2}^{2} \\ p_{4} = \frac{1}{2} m_{2} l_{1} l_{2} \end{matrix}

(47)

According to equation (2), the external resultant force without constraint can be obtained as²⁶

Q (q, \overset{\cdot}{q}, t) = - C (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G (q)

(48)

In equation (48), because the robot has not been constrained, the vector of applied joint torque is $τ = [0, 0]^{T}$ .

As shown in Figure 1, the position of the end-effector is obtained as

\begin{matrix} X_{1} = 0 \\ X_{2} = \sqrt{l_{1}^{2} + l_{2}^{2} - 2 l_{1} l_{2} \cos (2 q_{1})} \end{matrix}

(49)

where $[X_{1} X_{2}]$ is the coordinate of the end-effector in the global coordinate system.

Considering the end-effector of the robot is constrained with the vertical surface, one can get

\begin{matrix} l_{1} \cos q_{1} + l_{2} \cos (q_{1} + q_{2}) = X_{1} \\ l_{1} \sin q_{1} + l_{2} \sin (q_{1} + q_{2}) = X_{2} \end{matrix}

(50)

Taking time derivation twice on equation (50), one can obtain

\begin{matrix} - l_{1} \sin q_{1} {\overset{\cdot\cdot}{q}}_{1} - l_{2} \sin (q_{1} + q_{2}) {\overset{\cdot\cdot}{q}}_{1} - l_{2} \sin (q_{1} + q_{2}) {\overset{\cdot\cdot}{q}}_{2} \\ = l_{1} \cos q_{1} {\overset{\cdot}{q}}_{1}^{2} + l_{2} \cos (q_{1} + q_{2}) ({\overset{\cdot}{q}}_{1} + {\overset{\cdot}{q}}_{2})^{2} + {\overset{\cdot\cdot}{X}}_{1} \\ l_{1} \cos q_{1} {\overset{\cdot\cdot}{q}}_{1} + l_{2} \cos (q_{1} + q_{2}) {\overset{\cdot\cdot}{q}}_{1} + l_{2} \cos (q_{1} + q_{2}) {\overset{\cdot\cdot}{q}}_{2} \\ = l_{1} \sin q_{1} {\overset{\cdot}{q}}_{1}^{2} + l_{2} \sin (q_{1} + q_{2}) ({\overset{\cdot}{q}}_{1} + {\overset{\cdot}{q}}_{2})^{2} + {\overset{\cdot\cdot}{X}}_{2} \end{matrix}

(51)

Differentiating equation (49) with respect to time twice, the following equation is given by

\begin{matrix} {\overset{\cdot\cdot}{X}}_{1} = 0 \\ {\overset{\cdot\cdot}{X}}_{2} = \frac{2 n}{m} {\overset{\cdot\cdot}{q}}_{1} + \frac{4 m^{2} p - 4 n^{2}}{m^{3}} {\overset{\cdot}{q}}_{1}^{2} \end{matrix}

(52)

where the three parameters m, n, and p are functions of physical parameters

\begin{matrix} m = \sqrt{l_{1}^{2} + l_{2}^{2} - 2 l_{1} l_{2} \cos (2 q_{1})} \\ n = l_{1} l_{2} \sin (2 q_{1}) \\ p = l_{1} l_{2} \cos (2 q_{1}) \end{matrix}

(53)

Substituting equation (52) into equation (51), the second-order constraint equation²³ is shown as

A (q, \overset{\cdot}{q}, t) \overset{\cdot\cdot}{q} = b (q, \overset{\cdot}{q}, t)

(54)

in which

\begin{array}{l} A = [\begin{matrix} - l_{1} \sin q_{1} - l_{2} \sin (q_{1} + q_{2}) & - l_{2} \sin (q_{1} + q_{2}) \\ l_{1} \cos q_{1} + l_{2} \cos (q_{1} + q_{2}) - \frac{2 n}{m} & l_{2} \cos (q_{1} + q_{2}) \end{matrix}] \\ b = [\begin{matrix} l_{1} \cos q_{1} {\dot{q}}_{1}^{2} + l_{2} \cos (q_{1} + q_{2}) {({\dot{q}}_{1} + {\dot{q}}_{2})}^{2} \\ l_{1} \sin q_{1} {\dot{q}}_{1}^{2} + l_{2} \sin (q_{1} + q_{2}) {({\dot{q}}_{1} + {\dot{q}}_{2})}^{2} + \frac{4 m^{2} p - 4 n^{2}}{m^{3}} {\dot{q}}_{1}^{2} \end{matrix}] \end{array}

(55)

Now, M, A, b, and Q have been obtained, so the ideal constraint force $Q_{id}^{c}$ can be obtained according to equation (11).

As shown in Figure 1, the constraint function is

\begin{matrix} X_{1} = ϕ (x) = 0 \\ ϕ (q_{1}) = l_{1} \cos q_{1} + l_{2} \cos (q_{1} + q_{2}) = 0 \end{matrix}

(56)

Since the constraint equation is

Φ (q) = ϕ (h (q)) = ϕ (q_{1}) = l_{1} \cos q_{1} + l_{2} \cos (q_{1} + q_{2})

(57)

According to equation (18), the following is obtained

J (q, t) = \frac{\partial Φ (q)}{\partial q} = [\begin{matrix} - l_{1} \sin q_{1} - l_{2} \sin (q_{1} + q_{2}) & - l_{2} \sin (q_{1} + q_{2}) \end{matrix}]

(58)

Two situations to get the value of $λ$ are shown as follows:

Situation 1: When $q_{1} + q_{2} = π$ and $q_{1} = 0, q = π$ , we get $J (q, t) = 0$ . From Figure 1, one can easily find that two links of the robot are placed with overlapping each other. In this situation, $λ = 0$ is obtained, which means the constrained dynamic friction is zero.

Situation 2: When $q_{1} + q_{2} \neq π$ and $J (1) \neq 0, J (2) \neq 0$ , $λ$ can be obtained by equations (26) and (27).

In simulation, it is assumed that $l_{1} = l_{2} = 1, m_{1} = m_{2} = 1$ . According to equation (56), one can obtain

\cos q_{1} + \cos (q_{1} + q_{2}) = 0

(59)

The robot is constrained by the vertical surface, so $0 < q_{1} \leq π / 2, 0 \leq q_{2} < π$ . Then

\cos (q_{1} + q_{2}) = \cos (π - q_{1})

(60)

The relationship between $q_{1}$ and $q_{2}$ can be obtained as

q_{2} = π - 2 q_{1}

(61)

According to equation (61), one can obtain

L (q_{1}) = [\begin{matrix} 1 \\ - 2 \end{matrix}]

(62)

For the simulation, the reduced order model is used as the controlled object. The initial position is $q_{d 1} = 0.7 + 0.6 \cos t$ and the desired dynamic friction is $λ_{d} = 8 \sin t$ ; therefore, the initial value is chosen as $q = [\begin{matrix} 1.3 & π - 2.6 \end{matrix}]$ . The controller parameters are $K_{p} = [\begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}]$ , $K_{λ} = 8$ , and $Λ = 10.0$ . The simulation time is set to 20 s. The control scheme of the proposed control is shown in Figure 2.

Figure 2.

The control scheme of the controlled system.

Simulation results are shown in Figures 3 –8.

Figure 3.

The angle and angle speed tracking of the first link.

Figure 4.

The angle and angle speed tracking of the second link.

Figure 5.

Tracking errors of angles of the robot.

Figure 6.

Tracking errors of angle speed of the robot.

Figure 7.

The torque of the constrained robot.

Figure 8.

The tracking and tracking error of the dynamic friction.

Figures 3 and 4 show angles and angle speed tracking of the constrained robot, in which Figure 3 represents the first link and Figure 4 represents the second link. In Figure 3, solid red lines represent ideal values of $q_{1}$ and $d q_{1}$ and dashed blue lines represent practical values of $q_{1}$ and $d q_{1}$ . It can be found that solid lines are almost coincident with dashed lines. Similar to Figure 4, solid red lines represent ideal values of $q_{2}$ and $d q_{2}$ and dashed blue lines represent practical values of $q_{2}$ and $d q_{2}$ . Due to equation $2 q_{1} + q_{2} = π$ , as shown in Figures 3 and 4, the relationship between $q_{1}$ and $q_{2}$ has been validated obviously.

Figures 5 and 6 show tracking errors of angles and angle speed of the constrained robot system. It is obvious that tracking errors of angles and angle speed are almost equal to zero. In the stable region of Figures 5 and 6, the maximum error of $q_{1}$ and $q_{2}$ is less than 0.0005 rad and the maximum error of $d q_{1}$ and $d q_{2}$ is less than 0.001 rad/s. The main cause of this error and stability is the switch of discontinuous features in sliding mode control, which causes the chattering of the system. When the trajectory of the system gets to the switching surface, the moving point can pass through the switching surface forming the chattering. Major factors in the production of chattering mainly include four aspects: time delay switch, spatial delay switch, the inertia, and discreteness of the system.

Figure 7 represents the torque of the constrained robot, in which the solid red line represents torque values of the first link and the dashed blue line represents torque values of the second link. Because the second link is little affected by the static moment and the first link is affected by the static and dynamic moments, $τ_{1}$ is greater than the $τ_{2}$ .

Figure 8 shows the tracking and tracking error of the constrained dynamic friction, which can be seen as the non-ideal force in Udwadia and Kalaba theory. It is observed that the dashed red line represents desired values of $λ_{d}$ and the solid blue line represents practical values of $λ$ . It can be found that the tracking error between $λ$ and $λ_{d}$ is almost equal to zero.

From the above figures, these simulation results show that the dynamic model and the sliding mode control algorithm are achieved successfully.

Conclusion

In this article, a novel approach to the dynamic modeling and sliding mode control of the constrained robot system is proposed. By the Udwadia–Kalaba equation, expressions of the ideal and non-ideal forces are obtained and then the dynamic equation of the constrained robot system is established. Because the non-ideal constraint force is hardly calculated, the sliding mode control algorithm is presented for tracking the non-ideal force (the dynamic friction) in the constrained robot system. The major innovation in this article is the establishment of a new order reductive dynamic model and sliding mode controller to describe the constrained robot motion. Due to the lack of freedom, model order reduction method is creatively used in the Udwadia–Kalaba approach to complete the simulation of the constrained robot system. The model order reduction method can simplify the complexity of the calculation and can be extended to multi-degree of constrained robot system. A simple 2-DOF robot system with the vertical constraint is used to illustrate the methodology proposed in the article. From several simulation results, it can be found that the tracking errors are almost equal to zero. So, the proposed model and control method are feasible, correct, and valid.

Footnotes

Academic Editor: Elsa de Sa Caetano

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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