Kinematic Synthesis of Robotic Kinematic Synthesis of Robotic Manipulators from Task Manipulators from Task

DescriptionsDescriptions

June 2003June 2003By: Tarek Sobh, Daniel ToundykovBy: Tarek Sobh, Daniel Toundykov

Envisioning Optimal GeometryEnvisioning Optimal Geometry

Workspace Dimensionsand Coordinates of the Task-Points

Velocity and AccelerationRequirements

Obstacles, Working Medium, and Trajectory Biases

Restrictions on ManipulatorConfiguration

ObjectivesObjectives

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 TechniquesOptimization Techniques

Minimization of cost functionsMinimization of cost functionsStochastic algorithmsStochastic algorithmsParameters space methodsParameters space methodsCustom algorithms developed for specific Custom algorithms developed for specific types of robotstypes of robots

Steepest DescentSteepest Descent MethodMethod

System of equations is combined into a single function System of equations is combined into a single function whose zeroes correspond to the solution of the systemwhose zeroes correspond to the solution of the systemAlgorithm iteratively searches for local minima by Algorithm iteratively searches for local minima by investigating the gradient of the surface investigating the gradient of the surface S(x).S(x).Points where S(x) is small provide a good Points where S(x) is small provide a good approximation to the optimal solution.approximation to the optimal solution.

{{ffii(x)=0(x)=0} } → S(x)=∑→ S(x)=∑ffii(x)(x)22

Manipulability MeasureManipulability Measure

For performance purposes the For performance purposes the manipulability measure was the one manipulability measure was the one originally proposed by Tsuneo Yoshikawaoriginally proposed by Tsuneo YoshikawaSingular configurations are avoided by Singular configurations are avoided by maximizing the determinant of the maximizing the determinant of the Jacobian matrixJacobian matrix

w=w=√det(√det(J∙JJ∙JTT))

Optimization MeasureOptimization Measure

Optimization Measure

Task Points Manipulability Measure Dimensional Restrictions

Manipulator Jacobian

DOF & Types of Joints Joint Vector

Single Target ProblemSingle Target ProblemCost = [Cost = [b b + Manipulability]+ Manipulability]-1 -1 + + p p [Distance to target][Distance to target]

b b := bias to eliminate singularities:= bias to eliminate singularitiesp p := precision factor:= precision factor

Parameters that minimize the cost yield larger Parameters that minimize the cost yield larger manipulability and small positional errormanipulability and small positional errorIncrease of the precision factor forces the Increase of the precision factor forces the algorithm to reduce the positional error in order algorithm to reduce the positional error in order to compensate the overall cost growthto compensate the overall cost growth

Optimization for Multiple TargetsOptimization for Multiple Targets

Several single-target cost functions are Several single-target cost functions are combined into a single expressioncombined into a single expressionSingle-target cost functions share the Single-target cost functions share the same set of invariant DH-Parameters; same set of invariant DH-Parameters; however, each of these functions has its however, each of these functions has its own copy of the joint variablesown copy of the joint variables

Invariant DH-ParametersInvariant DH-Parameters

Invariant parameters depend on the types Invariant parameters depend on the types of jointsof jointsWhen no joints are specified, the algorithm When no joints are specified, the algorithm compares all possible configurations compares all possible configurations based on the average manipulability valuebased on the average manipulability valueInvariant DH-parameters have a dumping Invariant DH-parameters have a dumping factor. If dumping is large, the dimensions factor. If dumping is large, the dimensions of the robot must decrease to keep the of the robot must decrease to keep the total cost lowtotal cost low

Results of OptimizationResults of OptimizationSharedShared

DH-parameters DH-parameters

Joint VectorJoint Vectorfor Target 1for Target 1

……

Joint VectorJoint Vectorfor Target for Target NN

→→

→→

→→

Geometry that maximizes Geometry that maximizes manipulability at each manipulability at each

targettarget

Inverse SolutionInverse Solutionfor Target 1for Target 1

……

Inverse SolutionInverse Solution for Target for Target NN

MathematicaMathematica® ® (Wolfram Research Inc )(Wolfram Research Inc )

Powerful mathematical and graphics Powerful mathematical and graphics tools for scientific computingtools for scientific computingFlexible programming environmentFlexible programming environmentAvailability of enhancing technologies:Availability of enhancing technologies: Nexus to Java-based applications via Nexus to Java-based applications via

J/LinkJ/Link interface interface Flexible Web-integration provided by Flexible Web-integration provided by

webMathematicawebMathematica®® softwaresoftware Potential access to distributed computing Potential access to distributed computing

systems, such as systems, such as gridMatematicagridMatematica®®

CAD Module StructureCAD Module Structure

Computation Center

Generator ofJacobian Matrices

Generator ofTransformation

Matrices

Input Data Filter DynamicExpression Library

Graphics tools(use Rbotica package)

Generator ofOptimization Measure

File Processing Tools

Input DataInput Data

The set of task points The set of task points Configuration restrictions:Configuration restrictions: DOF value if the system should determine DOF value if the system should determine

optimal types of joints by itselfoptimal types of joints by itself or a specific configuration, such as Cartesian, or a specific configuration, such as Cartesian,

articulated etc.articulated etc.

Precision and size-dumping factorsPrecision and size-dumping factorsOutput file nameOutput file name

ScreenshotsScreenshots

Sample ISample I

Design a 3-link robot for a specific Design a 3-link robot for a specific parametric trajectoryparametric trajectoryNo configuration was given, so the No configuration was given, so the software had to choose the types of jointssoftware had to choose the types of jointsDimensions of the robot were severely Dimensions of the robot were severely restrictedrestricted

Sample I : TrajectorySample I : Trajectory

Sample I : DH-Table (PRP)Sample I : DH-Table (PRP)

LengthLength TwistTwist OffsetOffset AngleAngle

11 -0.61557 -0.61557 -0.0022699 -0.0022699 d1d1 0.037812 0.037812

22 -0.0025489 -0.0025489 1.56847 1.56847 5.0315 5.0315 xx1010-4-4 q2 q2

33 4.1630 4.1630 xx1010-4-4 0 0 d3 d3 0.92619 0.92619

Sample I : Manipulability EllipsoidsSample I : Manipulability Ellipsoids

Sample IISample II

The trajectory has been changedThe trajectory has been changedThis time we require a spherical This time we require a spherical manipulatormanipulatorNo significant spatial constraints have No significant spatial constraints have been providedbeen provided

Sample II : TrajectorySample II : Trajectory

Sample II : DH-Table (RRR)Sample II : DH-Table (RRR)

LengthLength TwistTwist OffsetOffset AngleAngle

11 1.6261 1.6261 -1.5700 -1.5700 -0.040365 -0.040365 q1q1

22 1.5632 1.5632 -4.9335 -4.9335 xx1010-4-4 -0.0012193 -0.0012193 q2 q2

33 1.5638 1.5638 0 0 1.8082 1.8082 xx1010-4-4 q3q3

Sample II : Manipulability EllipsoidsSample II : Manipulability Ellipsoids

Further ResearchFurther Research

Work has been done to account for robot Work has been done to account for robot dynamics and velocity requirementsdynamics and velocity requirementsOnline interface to the design moduleOnline interface to the design moduleFuture research may include obstacle Future research may include obstacle avoidance and integration with distributed avoidance and integration with distributed computing architecturescomputing architectures