Trajectory Control Study of Closed Kinematic Chain Mechanism

1
Trajectory Control Study of Closed Kinematic
Chain Mechanism

• Zhiyong Wang
• Department of Mechanical Engineering
• Rice University
• Houston, Texas
• E-mail wzy_at_rice.edu

2
Introduction Serial Robot vs. Parallel Robot

• Serial Robots
• Links are connected sequentially by revolute,
  prismatic or spherical joints.
• Typically, all the joints of serial robots are
  actuated.

• Parallel Robots
• Links are connected in series as well as in
  parallel combinations forming one or more
  closed-link loops.
• Typically, not all the joints of parallel robots
  are actuated.

3
Introduction Advantages of Parallel Robots

• Lighter moving parts, hence greater efficiency
  and faster acceleration at the end-effector.
• For closed chain mechanisms, the actuators are
  typically placed closer to the base or on the
  base itself.
• Greater rigidity to weight ratio
• Smaller positioning errors due to the
  multi-support of the end-effector.
• Greater payload handling capability for the same
  number of actuators.
• Applications
• Fast assembly lines
• Flight simulators
• Robotic machining.

4
Challenges In Modeling Control of
Closed-chain Mechanisms

• Unlike open-chain mechanisms, the derivation of
  dynamic equations of motion for closed-chain
  mechanisms suitable for controller design is
  still a active research area.
• Basic requirement of mode based controller
  design
• The system under study is described by a set of
  explicit ODEs, i.e. the number of differential
  equations is equal to degrees of freedom.
• Facts in modeling of closed-kinematic chain
  mechanisms
• The implicit nature of the loop-equations make
  it prohibitive to obtain explicit equations of
  motion for general close-chain mechanisms.
  Typically they are represented by system of DAEs.
• Domain of definition of generalized coordinates
  is a subset of the whole n-dimensional real space.

5
Derivation of a Reduced Model (I)

• Standard model for open-chain mechanisms
  explicit ODEs
• where n?n inertia matrix
• the n?n centrifugal and Coriolis terms
• the gravity vector
• applied torque at actuated joints
• Basic idea of model reduction

Free System ? Constrained System (ODEs)
(DAEs)
?
Constraints ? Lagrange Multiplier ?
6
Derivation of a Reduced Model (II)

• By considering the virtual work associated with
  the dynamics of the constrained system and
  assuming a workless constraint, we can eliminate
  the Lagrange Multipliers and obtain the equations
  of motion in terms of (q, q)
• where
• and

7
Derivation of a Reduced Model (III)

• Now the equations of motion of the constrained
  system can be expressed in terms of independent
  generalized coordinates as
• by combining
• where the parameterization q?(q) can be
  obtained from the constraint equations
  (loop-equations).
• The existence of ?(q) is insured by the implicit
  function theorem and it is in general implicit,
  therefore the Reduced Model is a implicit model.

8
Trajectory Control Implementation on the RPDR (I)

• Hardware environment
• Rice Planar Delta Robot (RPDR)

• dSPACE 1102 board
• Capable of computing the control law within 500
  us.
• ¼ hp Bodine 4435 DC motor with an output torque
  of 100 Oz-in for continuous operation at 2500 rpm
• Hengstler Optical Encoder with a resolution of
  10,000 pulses per revolution
• Links made of square, hollow steel tubing provide
  a high stiffness to weight ratio.

9
Trajectory Control Implementation on the RPDR (II)

• Trajectory Planning Inverse Kinematics.
• Selection of a singularity-free-region.
• Computation of desired position, velocity and
  acceleration.

10
Trajectory Control Implementation on the RPDR
(III)

• Real-time Computation of q?(q) .
• Rate of convergence.
• Simplicity of algorithm implementation.

Original formulation of Newton-Raphson iterative
algorithm for nonlinear algebraic
equation Intermittent initialization
algorithm Where is updated at a
lower frequency than the sampling frequency.
11
Trajectory Control Implementation on the RPDR (IV)

• Inverse dynamics control law
• where
• and

12
Trajectory Control Implementation on the RPDR (V)

• Experimental Results

Tracking Results with Initialization frequency
100 Hz
Tracking Results with Initialization frequency
1000 Hz
13
Trajectory Control Implementation on the RPDR (VI)

• Experimental Results (cont.)

Tracking Results with Initialization frequency 10
Hz
Tracking Results with Initialization frequency 20
Hz
14
Conclusion

• The trajectory control of RPDR is based on a
  implicit Reduced Model, thus the real-time
  computation of the parameterization q?(q) is a
  prerequisite to the implementation of the control
  law, which necessitate an effective numerical
  algorithm.
• In tracking control the effect of friction can
  not be ignored. More sophisticated model
  including friction shall improve the tracking
  results.
• The establishment of stability property of the
  inverse dynamics control law will be a useful
  extension of this work.