Robot Dynamics and Control
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This chapter presents an introduction to the dynamics and control of robot manipulators. We derive the equations of motion for a general open-chain manipulator and, using the structure present in the dynamics, construct control laws for asymptotic tracking of a desired trajectory. In deriving the dynamics, we will make explicit use of twists for representing the kinematics of the manipulator and explore the role that the kinematics play in the equations of motion. We assume some familiarity with dynamics and control of physical systems.
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Summary
The following are the key concepts covered in this chapter:
-
The equations of motion for a mechanical system with Lagrangian
satisfies Lagrange's equations:
where
is a set of generalized coordinates for the
system and 
represents the vector of
generalized external forces.
- The equations of motion for a rigid body with configuration
are given by the Newton-Euler equations:
where
is the mass of the body, 
is the inertia tensor, and
and 
represent the instantaneous body
velocity and applied body wrench.
- The equations of motion for an open-chain robot manipulator can be
written as
where
is the set of joint
variables for the robot and 
is the set of actuator forces applied at the joints. The dynamics of
a robot manipulator satisfy the following properties:
-
is symmetric and positive definite.
-
is a skew-symmetric matrix.
-
-
An equilibrium point
for the system 
is locally asymptotically stable if all solutions which start near
approach as 
. Stability can be checked using the
direct method of Lyapunov, by finding a locally positive
definite function 
such that 
is a
locally positive definite function along trajectories of the system.
In situations in which 
is only positive semi-definite,
Lasalle's invariance principle can be used to check asymptotic
stability. Alternatively, the indirect method of Lyapunov can
be employed by examining the linearization of the system, if it
exists. Global exponential stability of the linearization implies
local exponential stability of the full nonlinear system.
- Using the form and structure of the robot dynamics, several control
laws can be shown to track arbitrary trajectories. Two of the most
common are the computed torque control law,
and an augmented PD control law,
Both of these controllers result in exponential trajectory tracking of a given joint space trajectory. Workspace versions of these control laws can also be derived, allowing end-effector trajectories to be tracked without solving the inverse kinematics problem. Stability of these controllers can be verified using Lyapunov stability.
