Table of Contents generated with DocToc

• Gravity Compensation
  • DH Parameters
  • Simulation
  • Summary
  • MTM Dynamatics
    • Parallel Mechanism
  • Demo Video
    • Other Approaches
  • Appendix I: MTM DH Parameters
  • Appendix II: KDL with DH parameters
  • Appendix III: Matlab Toolbox

This page is deprecated

Please check README.md on https://github.com/jhu-dvrk/dvrk-gravity-compensation instead.

Gravity Compensation

In ideal case, given robot link mass and center of mass, Recursive Newton Euler (RNE) algorithm can be used to compute gravity compensation terms. To get this correct, we need two things:

• A correct implementation of RNE
• Robot model
  • Kinematics: DH Parameters
  • Dynamatics: Link mass & Center of Mass (COM)

In this section, we use a simple two link RR robot (RRBot in Gazebo tutorial) as a testbed for different robot kinematics and dynamics libraries in particular cisstRobot and KDL (See Appendix II). The robot parameters are known and described in an URDF file. Also MATLAB Robotics Vision & Control (RVC) toolbox is used as reference. Besides, we send computed values to Gazebo Simulator to visually check the values.

See:

• Gazebo tutorial: http://gazebosim.org/wiki/Tutorials/1.9/Using_A_URDF_In_Gazebo
• Code: https://github.com/ros-simulation/gazebo_ros_demos
• Code: https://github.com/ros-simulation/gazebo_ros_demos

DH Parameters

Table: Standard DH for RRBot

Frame	Joint Name	alpha	a	d	theta
1	Joint 1	0	0.9	0	q1
2	Joint 2	0	0.9	0.1	q2

Table: Modified DH for RRBot

Frame	Joint Name	alpha	a	d	theta
1	Joint 1	0	0.0	0	q1
2	Joint 2	0	0.9	0.1	q2
3	Tip	0	0.9	0	0

Simulation

A Gazebo model plugin has been written for testing.

Code Repository: https://github.com/zchen24/gazebo_ros_demos

Summary

CisstRobot & KDL have been tested in this simple case using both Standard and Modified DH parameters.

	Cisst Robot	KDL	Matlab
Kinematics Std	YES	YES	YES
Kinematics Mod	YES	YES	YES
Gravity Std	YES	YES	YES
Gravity Mod	NO	NO (? Not sure)	YES

NOTE:

1. center mass is with reference to link frame
2. use Standard DH for dynamics

MTM Dynamatics

NOTE:

• COM is with reference to link frame (Standard DH)
• Unit: m, kg

Link 7: Wrist Roll
NOTE: no mass, information is integrated to link 6

Link 6: Wrist Yaw

Link	Mass	COM	Comment
6	0.05	[0.0 -0.025 0.05]	Motor is heavy

Link 5: Wrist Pitch
NOTE: Left / Right are mirrored, different COM

Link	Mass	COM	Comment
5	0.04	[0.0 0.036 -0.065]	MTMR
5	0.04	[0.0 -0.036 -0.065]	MTML

NOTE: massive spring here, don't know how to deal with this.

Link 4: Setup Joint (Platform)
NOTE: Left / Right are mirrored, different COM

Link	Mass	COM	Comment
4	0.14	[0.0 -0.084 -0.12]	MTMR
4	0.14	[0.0 -0.084 0.12]	MTML

Link 3: Outer Pitch 2 (Elbow)

Link	Mass	COM	Comment
3	0.04	[-0.25 0.00 0.00]	Parallel Mechanism

taugc = torque computed using RNE  
tau(3) = taugc(3) - m * g * cos(q2 + q3)   // parallel 
if (q[3] < 0.05) tau[3] += ((q[3] - 0.05) * 0.1 - 0.07);  // cable

Parallel Mechanism

Link 2: Outer Pitch 1 (Shoulder)

Link	Mass	COM	Comment
3	0.65	[-0.38 0.00 0.00]

NOTE: huge mass at the top of this link, thus the COM is at the top.

tau[2] = taugc - 0.30    // 0.30 is offset for cable
if (q[2] < 0.05) tau[2] += (q[2] - 0.05) * 1.0 + 0.05   // cable 

Link 1: Outer Yaw

Link	Mass	COM	Comment
1	0.00	[0.00 0.00 0.00]

// Cable MTMR at JHU
tau[1] = -0.1 * qd[1]    // add damping
if (q[1] > -0.15) tau[1] = tau(1) + (q[1] - (-0.15)) * 0.1 + 0.04; 

NOTE:

• Disk set mass to 0, does not affect RNE computation
• Huge cable force

Demo Video

Other Approaches

See:
Atkeson, Christopher G., Chae H. An, and John M. Hollerbach. "Estimation of inertial parameters of manipulator loads and links." The International Journal of Robotics Research 5.3 (1986): 101-119.

??? Anyone wants to try ???

Appendix I: MTM DH Parameters

Standard DH Parameters

Image from Adnan Munawar (WPI)

Table 1: Standard DH for MTM

Frame	Joint Name	alpha	a	d	theta
1	Outer Yaw	pi/2	0	0	q1 - pi/2
2	Outer Pitch 1	0	l_arm	0	q2 - pi/2
3	Outer Pitch 2	-pi/2	l_forearm	0	q3 + pi/2
4	Setup Joint	pi/2	0	h	q4
5	Wrist Pitch	-pi/2	0	0	q5
6	Wrist Yaw	pi/2	0	0	q6 - pi/2
7	Wrist Roll	0	0	0	q7 + pi/2

---

Modified DH Parameters

Image from ISI da Vinci Research Kit Manual

Frame	Joint Name	alpha	a	d	theta
1	Outer Yaw	0	0	0	q1 + pi/2
2	Outer Pitch 1	-pi/2	0	0	q2 - pi/2
3	Outer Pitch 2	0	-l_arm	0	q3 + pi/2
4	Setup Joint	pi/2	-l_forearm	h	q4
5	Wrist Pitch	-pi/2	0	0	q5
6	Wrist Yaw	pi/2	0	0	q6 + pi/2
7	Wrist Roll	pi/2	0	0	q7 + pi/2

Appendix II: KDL with DH parameters

Kinematics and Dynamics Library (KDL) is a library that supports chain/tree like manipulator kinematics and dynamics computation. By default, it uses frame to represent adjacent joint/link relations, which is more flexiable. DH parameter is also supported as showed in the following code snippet.

Table: Standard DH for RRBot

Frame	Joint Name	alpha	a	d	theta
1	Joint 1	0	0.9	0	q1
2	Joint 2	0	0.9	0	q2

#include <kdl/chainfksolverpos_recursive.hpp>
#include <kdl/chainidsolver_recursive_newton_euler.hpp>

// Construct KDL 
KDL::Chain RRBotKdl;
inert = KDL::RigidBodyInertia(1.0, KDL::Vector(-0.45, 0, 0), 
                              KDL::RotationalInertia(1, 1, 1, 0, 0, 0));
RRBotKdl.addSegment(KDL::Segment(KDL::Joint(KDL::Joint::RotZ), 
                    KDL::Frame::DH(0.9, 0.0, 0.0, 0.0), inert));
RRBotKdl.addSegment(KDL::Segment(KDL::Joint(KDL::Joint::RotZ), 
                    KDL::Frame::DH(0.9, 0.0, 0.1, 0.0), inert));

// Get some joint pos, vel, acc values
KDL::JntArray jnt_q(mNumJnts);
KDL::JntArray jnt_qd(mNumJnts);
KDL::JntArray jnt_qdd(mNumJnts);
KDL::JntArray jnt_taugc(mNumJnts);
KDL::Wrenches jnt_wrenches;
for (unsigned int i = 0; i < mNumJnts; i++) {
  jnt_q(i) = q[i];
  jnt_qd(i) = 0.0;
  jnt_qdd(i) = 0.0;
  jnt_wrenches.push_back(KDL::Wrench());
}

// Kinematics 
KDL::ChainFkSolverPos_recursive fkSolver = KDL::ChainFkSolverPos_recursive(RRBotKdl);
KDL::Frame fkKDL;
fkSolver.JntToCart(jnt_q, fkKDL);

// Compute Dynamics 
KDL::Vector gravity(-9.81, 0.0, 0.0);
KDL::ChainIdSolver_RNE gcSolver = KDL::ChainIdSolver_RNE(RRBotKdl, gravity);
ret = gcSolver.CartToJnt(jnt_q, jnt_qd, jnt_qdd, jnt_wrenches,jnt_taugc);
if (ret < 0) ROS_ERROR("KDL: inverse dynamics ERROR");

NOTE: Support both standard & modified DH Wrenches

Reference:

• ROS Answers: http://answers.ros.org/question/9545/kdl-for-arm/
  • NOTE: Jon Bohren mentioned first joint should be fixed, don't understand why
• http://www.orocos.org/wiki/orocos/kdl-wiki

Appendix III: Matlab Toolbox

% start rvc toolbox
startup_rvc

% construct DH robot 

% L(1) 1st Revolute
L(1) = Link([0 0 0 pi/2 0]); 
L(1).offset = -pi/2;
L(1).m = 0.00;
L(1).r = [0 0 0];
L(1).I = [0.001, 0.001, 0.001, 0, 0, 0];
L(1).G = 1;
L(1).Jm = 0.0;

% L(2) 2nd Revolute
L(2) = Link([0 0 l_arm 0 0]); 
L(2).offset = -pi/2;
L(2).m = 0.10;
L(2).r = [-0.1794, 0, 0];
L(2).I = [0.001, 0.001, 0.001, 0, 0, 0];
L(2).G = 1;
L(2).Jm = 0.0;

% Create Serial Link
rob = SerialLink(L, 'name', 'Two link robot', ...
    'manufacturer', 'Zihan');


% Forward Kinematics
rob.fkine(q)

% Gravity 
rob.gravload(q)

NOTE: it also supports symbolic computation

List of codes:

• mdl_mtm.m: create MTM model with standard DH
• mdl_mtm_modified.m: create MTM model with modified DH
• mdl_psm.m: create PSM model with modified DH

Reference:

• Robotics, Vision and Control by Peter Corke
  • Chapter 5: kinematics
  • Chapter 7: dynamics
  • NOTE: more examples can be found in the book