kinematic synthesis of robotic manipulators from task descriptions
DESCRIPTION
Kinematic Synthesis of Robotic Manipulators from Task Descriptions. June 2003 By: Tarek Sobh, Daniel Toundykov. Envisioning Optimal Geometry. Objectives. Parameters considered in this work: Coordinates of the task-points Spatial constraints Restrictions (if any) on the types of joints - PowerPoint PPT PresentationTRANSCRIPT
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