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 Velocity and Acceleration Requirements Obstacles, Working Medium, and Trajectory Biases Restrictions on Manipulator Configuration

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 DescentMethod 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. {f i (x)=0} → S(x)=∑f i (x) 2

Manipulability Measure 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 w=√det(J∙J T )

Optimization Measure Task Points Manipulability Measure Manipulator Jacobian DOF & Types of Joints Joint Vector Dimensional Restrictions

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 SharedDH-parameters Joint Vector for Target 1 … Joint Vector for Target N → → → Geometry that maximizes manipulability at each target Inverse Solution for Target 1 … Inverse Solution for Target N 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 Nexus to Java-based applications via J/Link interface Flexible Web-integration provided by webMathematica ® software Flexible Web-integration provided by webMathematica ® software Potential access to distributed computing systems, such as gridMatematica ® Potential access to distributed computing systems, such as gridMatematica ®

CAD Module Structure Computation Center Generator of Jacobian Matrices Generator of Transformation Matrices Generator of Optimization Measure Input Data Filter Dynamic Expression Library Graphics tools (use Rbotica package) File Processing Tools

Input Data The set of task points Configuration restrictions: DOF value if the system should determine optimal types of joints by itself DOF value if the system should determine optimal types of joints by itself or a specific configuration, such as Cartesian, articulated etc. 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) LengthTwistOffsetAngle d x10 -4 q x d

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) LengthTwistOffsetAngle q x q x10 -4 q3

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