Kinematic Synthesis of Robotic
Manipulators from Task
Descriptions
June 2003
By: Tarek Sobh, Daniel Toundykov
Envisioning Optimal Geometry
Workspace Dimensions
and Coordinates of the Task-Points
Restrictions on Manipulator
Configuration
Velocity and Acceleration
Requirements
Obstacles, Working Medium, and
Trajectory Biases
Objectives
Parameters considered in this work:



Coordinates of the task-points
Spatial constraints
Restrictions (if any) on the types of joints
Goals



Simplified interface
Performance
Modular architecture to enable additional
optimization modules (for velocity, obstacles, etc.)
Optimization Techniques
Minimization of cost functions
Stochastic algorithms
Parameters space methods
Custom algorithms developed for specific
types of robots
Steepest Descent Method
{fi(x)=0} → S(x)=∑fi(x)2
System of equations is combined into a single function
whose zeroes correspond to the solution of the system
Algorithm iteratively searches for local minima by
investigating the gradient of the surface S(x).
Points where S(x) is small provide a good
approximation to the optimal solution.
Manipulability Measure
w=√det(J∙JT)
For performance purposes the
manipulability measure was the one
originally proposed by Tsuneo Yoshikawa
Singular configurations are avoided by
maximizing the determinant of the
Jacobian matrix
Optimization Measure
Optimization Measure
Task Points
Manipulability Measure
Dimensional Restrictions
Manipulator Jacobian
DOF & Types of Joints
Joint Vector
Single Target Problem
Cost = [b + Manipulability]-1 + p [Distance to target]
b := bias to eliminate singularities
p := precision factor
Parameters that minimize the cost yield larger
manipulability and small positional error
Increase of the precision factor forces the
algorithm to reduce the positional error in order
to compensate the overall cost growth
Optimization for Multiple Targets
Several single-target cost functions are
combined into a single expression
Single-target cost functions share the
same set of invariant DH-Parameters;
however, each of these functions has its
own copy of the joint variables
Invariant DH-Parameters
Invariant parameters depend on the types
of joints
When no joints are specified, the algorithm
compares all possible configurations
based on the average manipulability value
Invariant DH-parameters have a dumping
factor. If dumping is large, the dimensions
of the robot must decrease to keep the
total cost low
Results of Optimization
Shared
DH-parameters
→
Geometry that maximizes
manipulability at each
target
Joint Vector
for Target 1
→
Inverse Solution
for Target 1
…
Joint Vector
for Target N
…
→
Inverse Solution
for Target N
Mathematica®
(Wolfram Research Inc )
Powerful mathematical and graphics
tools for scientific computing
Flexible programming environment
Availability of enhancing technologies:



Nexus to Java-based applications via
J/Link interface
Flexible Web-integration provided by
webMathematica® software
Potential access to distributed computing
systems, such as gridMatematica®
CAD Module Structure
Computation Center
Input Data Filter
Dynamic
Expression Library
Graphics tools
(use Rbotica package)
File Processing Tools
Generator of
Jacobian Matrices
Generator of
Transformation
Matrices
Generator of
Optimization Measure
Input Data
The set of task points
Configuration restrictions:


DOF value if the system should determine
optimal types of joints by itself
or a specific configuration, such as Cartesian,
articulated etc.
Precision and size-dumping factors
Output file name
Screenshots
Sample I
Design a 3-link robot for a specific
parametric trajectory
No configuration was given, so the
software had to choose the types of joints
Dimensions of the robot were severely
restricted
Sample I : Trajectory
Sample I : DH-Table (PRP)
Length
Twist
Offset
Angle
-0.61557
-0.0022699
d1
0.037812
2 -0.0025489
1.56847
5.0315 x10-4
q2
3 4.1630 x10-4
0
d3
0.92619
1
Sample I : Manipulability Ellipsoids
Sample II
The trajectory has been changed
This time we require a spherical
manipulator
No significant spatial constraints have
been provided
Sample II : Trajectory
Sample II : DH-Table (RRR)
Length
Twist
Offset
Angle
1
1.6261
-1.5700
-0.040365
q1
2
1.5632
-4.9335 x10-4 -0.0012193
q2
3
1.5638
1.8082 x10-4
q3
0
Sample II : Manipulability Ellipsoids
Further Research
Work has been done to account for robot
dynamics and velocity requirements
Online interface to the design module
Future research may include obstacle
avoidance and integration with distributed
computing architectures