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Control of mechanical systems

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David A Roberts edited this page Dec 2, 2015 · 1 revision

A vertically-revolute type robot with two degrees-of-freedom. Mechanical systems as a controlled objective are mostly characteristic of multi-DOF (Degrees-Of-Freedom), existence of strong nonlinearities due to rotational joints, subjection to physical constraints, and redundancy in DOF. The desirable control goals may be related to not only physical variables of position and velocity of the system but also force or torque that is exerted on environment.

In nature, motion of mechanical systems is governed by the Lagrange equation that follows from the variational principle in mechanics, as described early on by Landau and Lifschitz (1960).

Robot motion control is typical of control of mechanical systems with multi degrees-of-freedom. Motion of a robotic arm with <math>n</math> joints shown in <figref>Verticallyrobo.jpg</figref> is governed by the Lagrange equation in terms of the vector of joint angle <math>q = (q_1, \cdots, q_n)^{\rm T}\ :</math>

<math>\label{eq-1}
H (q) \ddot{q} + \left\{ \frac{1}{2} \dot{H} (q) + S (q, \dot{q}) \right\} \dot{q} + g (q) = u </math>

where <math>\dot{q}</math> denotes the vector of joint angular velocities defined as <math>\dot{q} = {\rm d}q/ {\rm d}t\ ,</math> the derivative of <math>q</math> in time parameter <math>t\ ,</math> <math>\ddot{q} = {\rm d} \dot{q}/ {\rm d} t\ ,</math> <math>H (q) = (h_{ij} (q))</math> the <math>n \times n</math> inertia matrix, <math>g (q)</math> the gravity torque vector defined as a gradient of the gravity potential <math>P (q)\ ,</math> that is, <math>g (q) = \partial P (q)/ \partial q\ ,</math> <math>u</math> the external joint torque that can be regarded as control input, and <math>S (q, \dot{q}) = (s_{ij} (q))</math> is given by (e.g. Arimoto (1996))

<math>\label{eq-2}
s_{ij} (q) = \frac{1}{2} \left\{ \frac{\partial}{\partial q_j} \left( \sum_{k=1}^n \dot{q}_k h_{ik} (q) \right) - \frac{\partial}{\partial q_i} \left( \sum_{k=1}^n \dot{q}_k h_{jk} (q) \right) \right\} </math>

From this form, <math>S (q, \dot{q})</math> is homogeneous in <math>\dot{q}</math> and skew-symmetric, that is, <math>S^{\rm T} (q, \dot{q}) = - S (q, \dot{q})\ ,</math> and hence <math>\dot{q}^{\rm T} S (q, \dot{q}) \dot{q} = 0\ .</math> The inner product of eq. \eqref{eq-1} and <math>\dot{q}</math> leads to <math>\dot{q}^{\rm T} u = {\rm d} E (q, \dot{q})/ {\rm d} t\ ,</math> where <math>E (q, \dot{q}) = K (q, \dot{q}) + P (q), \ K (q, \dot{q}) = (1/2) \dot{q}^{\rm T} H (q) \dot{q}\ .</math> Here, <math>K</math> stands for the kinetic energy and <math>E</math> the total energy of the system. Then, the Lagrange equation of motion described by eq.\eqref{eq-1} follows from the variational principle applied to the Lagrangian: <math>L (q, \dot{q}) = K (q, \dot{q}) - P (q)\ .</math> Since the number of independent control inputs <math>n=\dim(u)</math> is equal to the number of DOF, such mechanical systems are said fully actuated.

Table of Contents

Passivity-based Control

Energy is one of the fundamental concepts in control of mechanical systems with multi-degrees-of-freedom. The action of a controller can be understood in energy terms as a dynamical system called "actuator" that supplies energies to the controlled system, upon interconnection, to modify desirably the behavior of the closed-loop (interconnected) system. This idea has its origin in Takegaki and Arimoto (1981) and is later called the "energy-shaping" approach, which is now known as a basic controller design technic common in control of mechanical systems. Its systematic interpretation is called "passivity-based control".

Given a target posture <math>q_d</math> for a robot manipulator, consider the two control inputs:

<math>\label{eq-3}
{\rm a)}\quad u = g (q) - C \dot{q} - A (q - q_d), \qquad {\rm b)} \quad u = g (q_d) - C \dot{q} - A (q - q_d) </math>

where <math>C</math> and <math>A</math> denote an <math>n \times n</math> constant gain matrix with positive definiteness. This signal form is called a "PD (Position and Derivative) feedback" with gravity compensation. It presumes not only real-time measurements of joint angles <math>q_i</math> and angular velocities <math>\dot{q}_i</math> but also real-time computation of the gravity function <math>g (q)</math> in the case of a). Gravity compensation in the case of b) is regarded as a target torque regulation that enough withstands the external joint torques to be exerted from the gravitational force at its target position. Substitution of the control signal b) of eq.\eqref{eq-3} into eq.\eqref{eq-1} yields

<math>\label{eq-4}
H (q) \ddot{q} + \left\{ \frac{1}{2} \dot{H} (q) + S (q, \dot{q}) + C \right\} \dot{q} + g (q) - g (q_d) + A (q - q_d) = 0 </math>

that is called the closed-loop dynamics. The position control problem for the system with a specified target posture <math>q = q_d</math> is now interpreted in terms of the mathematical method of Lyapunov's stability, that is, to prove the theorem that any solution trajectory (that is called "orbit") to eq.\eqref{eq-4} starting from a neighborhood of the equilibrium state <math>(q_d, \dot{q} = 0)</math> remains in its vicinity and converges asymptotically to it as <math>t \to \infty\ .</math> This stability proof can be established by finding Lyapunov's relation by taking the inner product of eq.\eqref{eq-4} and <math>\dot{q}\ ,</math> which results in

<math></math>\label{eq-5}
\frac{\rm d}
<math>\label{eq-13}
u = - A_1 \Delta q - B_1 \Delta \dot{q} + Y (q_d, \dot{q}_d, \dot{q}_d, \ddot{q}_d) \hat{\Theta} </math>

where <math>\hat{\Theta}</math> is continuously updated by

<math>\label{eq-14}
\hat{\Theta} (t) = \hat{\Theta} (0) - \int_0^t \Gamma^{-1} Y^{\rm T} \left( q_d (\tau), \dot{q}_d (\tau), \dot{q}_d (\tau), \ddot{q}_d (\tau) \right) y (\tau) {\rm d} \tau </math>

where <math>\Delta q = q - q_d\ ,</math> <math>\Delta \dot{q} = \dot{q} - \dot{q}_d\ ,</math> and <math>y = \Delta \dot{q} + \gamma \Delta q\ .</math> This method of adaptive scheme was originally proposed by Slotine and Li (1987) and now called the model-based adaptive control.

References

  • Landau, L.D., and Lifschitz, E.M. (1960) Mechanics: Vol.1 of Course of Theoretical Physics, The third edition in 1976, Elsevier, Amsterdam, The Netherlands.
  • Arimoto, S. (1996) Control Theory of Nonlinear Mechanical Systems: A Passivity-based and Circuit-theoretic Approach, Oxford Univ. Press, Oxford, UK.
  • Takegaki, M., and Arimoto, S. (1981) A new feedback method for dynamic control of manipulators, ASME J. of Dyn. Syst. Meas. and Control, 103:119.
  • McClamroch, N.H., and Wang, D. (1990) Linear feedback control of position and contact force for a nonlinear constrained mechanism, ASME J. of Dyn. Syst. Meas. and Control, 112:640.
  • Wang, D., and McClamroch, N.H. (1993) Position and force control for constrained manipulator motion: Lyapunov's direct method, IEEE Trans. Rob. Autom., 9:308.
  • Bernstein, N.A. (1996) (translated from the Russian by M.L. Latash) On Dexterity and its Development, Lawrence Erlbaum Associates, Inc. USA.
  • Arimoto, S. (1995) Fundamental problems of robot control: Part I. Innovation in the realm of robot servo-loops, Robotica, 13:19.
  • Hogan, N. (1985) The mechanics of multi-joint posture and movement control, Biol. Cybern., 52:315.
  • Thelen, E., and Smith, L.B. (1996) A Dynamic Systems Approach to the Development of Cognition and Action, MIT Press, Cambridge, MA, USA.
  • Hollerbach, J.M., and Suh, K.C. (1987) Redundancy resolution of manipulators through torque optimization, IEEE J. Rob. Autom., 3:308.
  • Arimoto, S., Hashiguch, H., Sekimoto, M., and Ozawa, R. (2005) Generation of natural motions for redundant multi-joint systems: A differential-geometric approach based upon the principle of least actions, J. of Robotic Systems. 22:583.
  • Seraji, H. (1989) Configuration control of redundant manipulators: Theory and implementation, IEEE Trans. Rob. Autom., 5:472.
  • Slotine, J.J.E., and Li, W. (1987) On the adaptive control of robot manipulators, Int. J. of Robotics Research, 6:49.
Internal references
  • James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
  • Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

Some Other Important Subjects on Control of Mechanical Systems

  • Control of Under-actuated Systems
  • Mechanical Systems under Nonholonomic Constraints
  • Control of Multi-fingered Hands (as a System with Multiple Contacts)

See also

Control theory

Category:Control Theory

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