Motion Control

Abstract

Robot manipulators have been widely used in industrial automation. In many modern robot control applications, sensory information such as visual feedback is used to improve positioning accuracy and robustness to uncertainty. This chapter introduces basic concepts and design methods that are employed for motion control of robot manipulators with uncertainty. The chapter covers both basic methods in joint-space control and advance topics in sensory task-space control.

Appendices

Appendix 1

Preliminaries on control theories:

Consider the following nonlinear system:

$$ \dot{x}=f\left(x,t\right) $$

(109)

where x ∈ R ⁿ is a vector of state of the system and f (.) is a nonlinear function.

Definition A1

The equilibrium points of the system Eq. 109 are defined as the state vectors x _e of x for which if at specific time t ₀, x = x _e, then x will remain unchanged for all t > t ₀. In other words, at equilibrium points the state of the system satisfies f (x _e ) = 0.

It is often important to know whether the equilibrium point is stable or not. In the following, a definition of stable equilibrium point is put forward:

Remark A1

It can be always assumed that the equilibrium point is zero by using change of variable y = x – x _e .

Definition A2

The equilibrium point x _e = 0 of the system Eq. 109 is said to be stable if for any ε > 0, there exists δ > 0 such that if ‖x(0) − 0‖ < δ, then ‖x(t) − 0‖ < ε for all t > 0. It can be mathematically represents as follows:

$$ \forall \varepsilon >0\kern0.36em \exists \delta >0,\kern0.36em if\kern0.24em \left\Vert x(0)-0\right\Vert <\delta \Rightarrow \left\Vert x(t)-0\right\Vert <\varepsilon \kern0.24em for\kern0.24em t>{t}_0. $$

(110)

To examine the stability of the equilibrium point, the Lyapunov theory can be utilized. The main advantage of the Lyapunov theory is that the stability of the system can be determined without solving the differential equations of the system. Moreover, the Lyapunov theory can be used to design controllers that stabilize nonlinear systems.

Before presenting the Lyapunov theory, a certain class of functions is introduced as follows:

Definition A3

A continuous function V : R ⁿ × R ₊ → R is a locally positive definite function (lpdf) if for some ε > 0 and some continuous, the following conditions hold:

$$ \left\{\begin{array}{l}I:V\left(0,t\right)=0\\ {}II:V\left(x,t\right)>0\kern1em \forall x\in {B}_{\varepsilon },\forall t\ge 0\end{array}\right. $$

(111)

where B _ε is a ball of size ε around the origin which is mathematically expressed as B _ε = {x ∈ R ⁿ : ‖x‖ < ε}.

In addition, V is a positive definite function (pdf) if the condition (II) is true for all x ∈ R.

If in condition (II) V (x, t) ≥ 0, then V (x, t) is a (locally) positive semi-definite function.

Remark A2

If V (x) = x ^T Mx, where M is a real symmetric matrix, then V is a pdf if and only if M is a positive definite matrix.

Theorem A1

Lyapunov stability theorem: Suppose x _e = 0 is an equilibrium point of the system Eq. 109. Let V(x, t) be a nonnegative function with derivative $\dot{V}$ V along trajectories of the system dynamics:

$$ \frac{dV}{dt}=\frac{\partial V}{\partial t}\dot{x}=\frac{\partial V}{\partial t}f\left(x,t\right). $$

(112)

• (i) If V (x, t) is a locally positive definite function and $\dot{V}$ is a locally positive semi-definite function in x and for all t, then the origin of the system is locally stable.

• (ii) If V (x, t) is a locally positive definite function and $\dot{V}$ is a locally positive definite function in x and for all t, then the origin of the system is locally asymptotically stable.

• (iii) If V (x, t) is a positive definite function and $\dot{V}$ is a positive definite function in x and for all t, then the origin of the system is globally asymptotically stable.

If the function V (x, t) exists in the above theorem, then it is called a Lyapunov function.

Remark A3

The Lyapunov stability theorem only provides sufficient conditions for the stability of nonlinear systems; hence, the failure of finding a Lyapunov function does not prove the instability of the nonlinear system

In the case that $-\dot{V}\left(x,t\right)$ is a positive semi-definite function, the Lyapunov stability theorem cannot provide any information on the asymptotic stability of the system. To deal with the stability of nonlinear autonomous systems when $-\dot{V}(x)$ is a positive semi-definite function, LaSalle’s invariance principle has been presented.

Lemma A1

LaSalle ’ s invariance principle : Let V : R ⁿ → R be a positive definite function such that $\dot{V}(x)\le 0$ in compact set Ω. Let D be the set of all points in Ω where $\dot{V}(x)=0$. Therefore, every solution of the system $\dot{x}=f(x)$ starting in Ω approaches to the largest invariant set inside D. In particular, if D contains no trajectories other than x = 0, then 0 is locally asymptotically stable.

LaSalle’s invariance principle enables one to conclude asymptotic stability only for autonomous systems. For non-autonomous systems, Barbalat’s lemma can be used.

Lemma A2

Barbalat ’ s lemma : If a function V (t, x) satisfies the following conditions:

• (i) V(x, t) is lower bounded.

• (ii) $\dot{V}\left(x,t\right)$ is negative semi-definite.

• (iii) $\dot{V}\left(x,t\right)$ is uniformly continuous in time or equivalently $\ddot{V}\left(t,x\right)$ is bounded. Then, $\dot{V}\left(x,t\right)$ goes to zero as t → ∞.

Appendix 2

Parameters of dynamic Eq. 4 of the two-link robot manipulator which is depicted in Fig. 1:

Elements of inertia matrix M(q) are

$$ \begin{array}{l}{M}_{11}=\frac{4}{3}{M}_1{L}_{c1}^2+\frac{4}{3}{M}_2{L}_{c2}^2+{M}_2{L}_1^2+2{M}_2{L}_1{L}_{c2} \cos \left({q}_2\right)\\ {}{M}_{12}=\frac{4}{3}{M}_2{L}_{c2}^2+{M}_2{L}_1{L}_{c2} \cos \left({q}_2\right)\\ {}{M}_{21}=\frac{4}{3}{M}_2{L}_{c2}^2+{M}_2{L}_1{L}_{c2} \cos \left({q}_2\right)\\ {}{M}_{22}=\frac{4}{3}{M}_2{L}_{c2}^2.\end{array} $$

(113)

Elements of matrix $C\left(q,\dot{q}\right)$ are given as

$$ \begin{array}{l}\kern-0.2em {C}_{11}=-{M}_2{L}_1{L}_{c2}{\dot{q}}_2 \sin \left({q}_2\right)\\ {}{C}_{12}=-{M}_2{L}_1{L}_{c2}\left[{\dot{q}}_1+{\dot{q}}_2\right] \sin \left({q}_2\right)\\ {}{C}_{21}={M}_2{L}_1{L}_{c2}{\dot{q}}_1 \sin \left({q}_2\right)\\ {}{C}_{22}=0.\end{array} $$

(114)

Elements of gravitational force matrix are

$$ \begin{array}{l}{g}_1=\left({M}_1{L}_{c1}+{M}_2{L}_1\right)\mathfrak{g} \cos \left({q}_1\right)+{M}_2{L}_{c2}\mathfrak{g} \cos \left({q}_1+{q}_2\right)\\ {}{g}_2={M}_2{L}_{c2}\mathfrak{g} \cos \left({q}_1+{q}_2\right)\end{array} $$

(115)

where $\mathfrak{g}$ is the gravity due to acceleration.

Appendix 3

Part of the proof of the task-space control law Eq. 55:

Differentiating the Lyapunov candidate V (expressed in Eq. 58) with respect to time gives

$$ \begin{array}{c}\dot{V}={\dot{q}}^TM(q)\ddot{q}+\frac{1}{2}{\dot{q}}^T\dot{M}(q)\dot{q}+\alpha {\ddot{q}}^TM(q){\widehat{J}}^{\dagger }(q)s\left(\tilde{x}\right)\\ {}+\alpha {\dot{q}}^T\dot{M}(q){\widehat{J}}^{\dagger }(q)s\left(\tilde{x}\right)+\alpha {\dot{q}}^TM(q){\dot{\widehat{J}}}^{\dagger }(q)s\left(\tilde{x}\right)\\ {}+\alpha {\dot{q}}^TM(q){\widehat{J}}^{\dagger }(q)\dot{s}\left(\tilde{x}\right)+{\dot{x}}^T{K}_ps\left(\tilde{x}\right)+\alpha {\dot{x}}^T{K}_vs\left(\tilde{x}\right).\end{array} $$

(116)

Substituting the closed-loop Eq. 57 into Eq. 116, using property 2 and simplifying, yields

$$ \begin{array}{c}\dot{V}=-\kern0.3em {\dot{q}}^TD(q)\dot{q}-{\dot{q}}^T{\widehat{J}}^T(q)\left({K}_ps\left(\tilde{x}\right)+{K}_v\dot{x}\right)\\ {}-\kern0.3em \alpha {\dot{q}}^T\left\{C\left(q,\dot{q}\right)+D(q)-\dot{M}(q)\right\}{\widehat{J}}^{\dagger }(q)s\left(\tilde{x}\right)\\ {}-\kern0.3em \alpha s{\left(\tilde{x}\right)}^T{K}_ps\left(\tilde{x}\right)+\alpha {\dot{q}}^TM(q){\dot{\widehat{J}}}^{\dagger }(q)s\left(\tilde{x}\right)\\ {}+\kern0.3em \alpha {\dot{q}}^TM(q){\widehat{J}}^{\dagger }(q)\dot{s}\left(\tilde{x}\right)+{\dot{x}}^T{K}_ps\left(\tilde{x}\right).\end{array} $$

(117)

Using Eq. 14, $\dot{V}$ can be written as follows:

$$ \begin{array}{c}\dot{V}=-{\dot{q}}^T\left\{{\widehat{J}}^T(q){K}_vJ(q)+D(q)\right\}\dot{q}-{\dot{q}}^T\left\{{\widehat{J}}^T(q)-{J}^T(q)\right\}{K}_ps\left(\tilde{x}\right)\\ {}\kern0.96em -\alpha {\dot{q}}^T\left\{C\left(q,\dot{q}\right)+D(q)-\dot{M}(q)\right\}{\widehat{J}}^{\dagger }(q)s\left(\tilde{x}\right)-\alpha s{\left(\tilde{x}\right)}^T{K}_ps\left(\tilde{x}\right)\\ {}\kern0.96em +\alpha {\dot{q}}^TM(q){\dot{\widehat{J}}}^{\dagger }(q)s\left(\tilde{x}\right)+\alpha {\dot{q}}^TM(q){\widehat{J}}^{\dagger }(q)\dot{s}\left(\tilde{x}\right).\end{array} $$

(118)

Since $s\left(\tilde{x}\right)$ is bounded, there exist constants c ₀ > 0 and c ₁ > 0 so that

$$ \begin{array}{c}-\alpha {\dot{q}}^T\left\{C\left(q,\dot{q}\right)+D(q)-\dot{M}(q)\right\}{\widehat{J}}^{\dagger }(q)s\left(\tilde{x}\right)+\alpha {\dot{q}}^TM(q){\dot{\widehat{J}}}^{\dagger }(q)s\left(\tilde{x}\right)\\ {}+\alpha {\dot{q}}^TM(q){\widehat{J}}^{\dagger }(q)\dot{s}\left(\tilde{x}\right)\le \alpha {c}_0{\left\Vert \dot{q}\right\Vert}^2+\alpha {c}_1{\left\Vert s\left(\tilde{x}\right)\right\Vert}^2.\end{array} $$

(119)

Substituting inequality Eq. 119 into Eq. 118 yields

$$ \begin{array}{c}\dot{V}\kern0.48em \le \kern0.6em -{\dot{q}}^T\left\{{\widehat{J}}^T(q){K}_vJ(q)+D(q)-\alpha {c}_0I\right\}\dot{q}-\alpha s{\left(\tilde{x}\right)}^T\left({K}_p-{c}_1I\right)s\left(\tilde{x}\right)\\ {}-{\dot{q}}^T\left\{{\widehat{J}}^T(q)-{J}^T(q)\right\}{K}_ps\left(\tilde{x}\right).\end{array} $$

(120)

Let $\tilde{J}(q)$ be the Jacobian estimation error which is defined as $\tilde{J}(q)=J(q)-\widehat{J}(q)$. Hence, Eq. 120 can be written with respect to J(q) and $\tilde{J}(q)$ as follows:

$$ \begin{array}{c}\dot{V}\kern0.48em \le \kern0.6em -{\dot{q}}^T\left\{{J}^T(q){K}_vJ(q)+D(q)-\alpha {c}_0I\right\}\dot{q}-\alpha s{\left(\tilde{x}\right)}^T\left({K}_p-{c}_1I\right)s\left(\tilde{x}\right)\\ {}+{\dot{q}}^T{\tilde{J}}^T(q){K}_vJ(q)\dot{q}+{\dot{q}}^T{\tilde{J}}^T(q){K}_ps\left(\tilde{x}\right).\end{array} $$

(121)

Since the Jacobian matrix contains trigonometric functions of q, ‖J(q)‖ ≤ b _J and Eq. 121 can be rewritten as

$$ \begin{array}{c}\dot{V}\le \kern0.36em -\left\{{\lambda}_{\min}\left({J}^T(q){K}_vJ(q)+D(q)\right)-\gamma {b}_J{\lambda}_{\max}\left({K}_v\right)-\alpha {c}_0\right\}{\left\Vert \dot{q}\right\Vert}^2\\ {}\kern1.32em +\gamma {\lambda}_{\max}\left({K}_p\right)\left\Vert s\left(\tilde{x}\right)\right\Vert \left\Vert \dot{q}\right\Vert -\alpha \left({\lambda}_{\min}\left({K}_p\right)-{c}_1\right){\left\Vert s\left(\tilde{x}\right)\right\Vert}^2\end{array} $$

(122)

where γ is defined in Eq. 56. Since

$$ 2\left\Vert s\left(\tilde{x}\right)\right\Vert \left\Vert \dot{q}\right\Vert \le {\left\Vert s\left(\tilde{x}\right)\right\Vert}^2+\kern0.36em {\left\Vert \dot{q}\right\Vert}^2, $$

(123)

therefore, Eq. 122 is simplified as follows:

$$ \begin{array}{c}\dot{V}\kern0.36em \le \kern0.36em -\left\{{\lambda}_{\min}\left({J}^T(q){K}_vJ(q)+D(q)\right)-\gamma {b}_J{\lambda}_{\max}\left({K}_v\right)-\frac{1}{2}\gamma {\lambda}_{\max}\left({K}_p\right)-\alpha {c}_0\right\}{\left\Vert \dot{q}\right\Vert}^2\\ {}\kern1.44em -\left(\alpha {\lambda}_{\min}\left({K}_p\right)-\frac{1}{2}\gamma {\lambda}_{\max}\left({K}_p\right)-\alpha {c}_1\right){\left\Vert s\left(\tilde{x}\right)\right\Vert}^2.\end{array} $$

(124)

Equation 124 can be written as follows:

$$ \begin{array}{c}\dot{V}\kern0.36em \le \kern0.48em -\left\{{\lambda}_{\max}\left({K}_v\right)\left({\delta}_1-\frac{\gamma }{2}\left({\delta}_2+2{b}_J\right)\right)-\alpha {c}_0\right\}{\left\Vert \dot{q}\right\Vert}^2\\ {}\kern1.56em -\left({\lambda}_{\max}\left({K}_p\right)\left(\alpha {\delta}_3-\frac{\gamma }{2}\right)-\alpha {c}_1\right){\left\Vert s\left(\tilde{x}\right)\right\Vert}^2\end{array} $$

(125)

such that

$$ \begin{array}{l}{\delta}_1=\frac{\lambda_{\min}\left[{J}^T(q){K}_vJ(q)+D(q)\right]}{\lambda_{\max}\left({K}_v\right)}\\ {}{\delta}_2=\frac{\lambda_{\max}\left({K}_p\right)}{\lambda_{\max}\left({K}_v\right)}\\ {}{\delta}_3=\frac{\lambda_{\min}\left({K}_p\right)}{\lambda_{\max}\left({K}_p\right)}.\end{array} $$

(126)