Module: robotics.special

additional support functions for robotics; The library is built on Denavit-Hartenberg Parameters and Homogeneous Transformations (HT) to describe transformations and coordinate systems

  • Author: Martin Sereinig

  • Date: 2021-22-09

Function: VelocityManipulability

VelocityManipulability(robot, HT, mode)

  • function description:
    compute velocity manipulability measure for given pose (homogeneous transformation)
  • input:
    robot: robot class
    HT: actual pose as homogeneous transformaton matrix
    mode: rotational or translational part of the movement
  • output:
    velocity manipulability measure as scalar value, defined as \(\sqrt(det(JJ^T))\)
  • author:
    Martin Sereinig
  • notes:
    compute velocity dependent manipulability definded by Yoshikawa, see

Function: ForceManipulability

ForceManipulability(robot, HT, mode, singularWeight = 100)

  • function description:
    compute force manipulability measure for given pose (homogeneous transformation)
  • input:
    robot: robot class
    HT: actual pose as hoogenious transformaton matrix
    singularWeight: Weighting of singular configurations where the value would be infinity, default value=100
    mode: rotational or translational part of the movement
  • output:
    force manipulability measure as scalar value, defined as \(\sqrt((det(JJ^T))^{-1})\)
  • author:
    Martin Sereinig
  • notes:
    compute force dependent manipulability definded by Yoshikawa, see

Function: StiffnessManipulability

StiffnessManipulability(robot, JointStiffness, HT, mode, singularWeight = 1000)

  • function description:
    compute cartesian stiffness measure for given pose (homogeneous transformation)
  • input:
    robot: robot class
    JointStiffness: joint stiffness matrix
    HT: actual pose as homogeneous transformaton matrix
    mode: rotational or translational part of the movement
    singularWeight: Weighting of singular configurations where the value would be infinity,default value=1000
  • output:
    stiffness manipulability measure as scalar value, defined as minimum Eigenvalaue of the Cartesian stiffness matrix
    Cartesian stiffness matrix
  • author:
    Martin Sereinig
  • notes:

  • status:
    this function is currently under development and under testing!

Function: JointJacobian

JointJacobian(robot, HTJoint, HTLink)

  • function description:
    compute joint jacobian for each frame for given pose (homogeneous transformation)
  • input:
    robot: robot class
    HT: actual pose as homogeneous transformaton matrix
  • output:
    Link(body)-Jacobi matrix JJ: \(\LU{i}{JJ_i}=[\LU{i}{J_{Ri}},\; \LU{i}{J_{Ti}}]\) for each link i, seperated in rotational (\(J_R\)) and translational (\(J_T\)) part of Jacobian matrix located in the \(i^{th}\) coordiante system, see
  • author:
    Martin Sereinig
  • notes:
    runs over number of HTs given in HT (may be less than number of links), caclulations in link coordinate system located at the end of each link regarding Standard Denavid-Hartenberg parameters, see

Function: MassMatrix

MassMatrix(robot, HT, jointJacobian)

  • function description:
    compute mass matrix from jointJacobian
  • input:
    robot: robot structure
    HT: actual pose as homogeneous transformaton matrix
    jointJacobian: provide list of jacobians as provided by function JointJacobian(…)
  • output:
    MM: Mass matrix
  • author:
    Martin Sereinig
  • notes:
    Mass Matrix calculation calculated in joint coordinates regarding (std) DH parameter:
    ** Dynamic equations in minimal coordinates as described in Mehrkörpersysteme by Woernle, , p206, eq6.90.
    ** Caclulations in link coordinate system at the end of each link

Relevant Examples (Ex) and TestModels (TM) with weblink to github:

Function: DynamicManipulability

DynamicManipulability(robot, HT, MassMatrix, Tmax, mode, singularWeight = 1000)

  • function description:
    compute dynamic manipulability measure for given pose (homogeneous transformation)
  • input:
    robot: robot structure
    HT: actual pose as homogeneous transformaton matrix
    Tmax: maximum joint torques
    mode: rotational or translational part of the movement
    MassMatrix: Mass (inertia) Maxtrix provided by the function MassMatrix
    singularWeight: Weighting of singular configurations where the value would be infinity,default value=1000
  • output:
    dynamic manipulability measure as scalar value, defined as minimum Eigenvalaue of the dynamic manipulability matrix N
    dynamic manipulability matrix
  • author:
    Martin Sereinig
  • notes:
    acceleration dependent manipulability definded by Chiacchio, see , eq.32. The eigenvectors and eigenvalues of N ([eigenvec eigenval]=eig(N))gives the direction and value of minimal and maximal accaleration )
  • status:
    this function is currently under development and under testing!

Function: CalculateAllMeasures

CalculateAllMeasures(robot, robotDic, q, mode, flag = [0,0,0,0])

  • function description:
    calculation of 4 different manipulability measures using a certain serial robot
  • input:
    robot: robot class
    robotDic: robot dictionary
    q: joint position vector
    mode: trans or rot, for used parts of the manipulator Jacobi Matrix
    Tmax: maximum joint torques
    mode: rotational or translational part of the movement
    flag: flag vector to swich individual measure on and of [flagmv,flagmf,flagmst,flagma] = [1,1,1,1]
  • output:
    [mv,mf,mst,mstM,ma,maM]
  • author:
    Martin Sereinig
  • notes:

  • status:
    this function is currently under development and under testing!