massMatrix

Joint-space mass matrix

Syntax

H = massMatrix(robot)

H = massMatrix(robot,configuration)

Description

H = massMatrix(robot) returns the joint-space mass matrix of the home configuration of a robot.

H = massMatrix(robot,configuration) returns the mass matrix for a specified robot configuration.

Examples

Calculate The Mass Matrix For A Robot Configuration

Load a model of the Boston Dynamics Atlas® humanoid robot from the Robotics System Toolbox™ loadrobot, returned as a rigidBodyTree object. Ensure that the data format is set to "row". For all dynamics calculations, the data format must be either "row" or "column".

atlas = loadrobot("atlas", DataFormat="row", Gravity=[0 0 -9.81]);

Generate a random configuration for atlas.

q = randomConfiguration(atlas);

Get the mass matrix at configuration q.

H = massMatrix(atlas,q)

H = 30×30

    5.9158   -1.2431    0.6367    0.4879   -0.5435    0.1373    0.1765   -0.0007   -0.0072   -0.0003   -0.0001    0.9609    0.3663   -0.0521   -0.3232   -0.0269    0.0011    0.0001         0         0         0         0         0         0         0         0         0         0         0         0
   -1.2431   20.9002    0.5109   -0.5926    0.8585    0.1457   -0.0027   -0.0003    0.0018   -0.0002   -0.0018   -0.0413    0.2023    0.0525    0.1088    0.0236    0.0108   -0.0005         0         0         0         0         0         0         0         0         0         0         0         0
    0.6367    0.5109   19.4204    0.2148    0.6959   -0.3081   -0.2676    0.0004    0.0074    0.0004    0.0000    0.0330    0.9184    0.0043   -0.3224   -0.0142    0.0250    0.0002         0         0         0         0         0         0         0         0         0         0         0         0
    0.4879   -0.5926    0.2148    0.2945   -0.0589    0.0975    0.0317   -0.0013    0.0032   -0.0003         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.5435    0.8585    0.6959   -0.0589    0.5987   -0.0512    0.0887    0.0014   -0.0112    0.0002         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
    0.1373    0.1457   -0.3081    0.0975   -0.0512    0.2033   -0.0008   -0.0010    0.0090   -0.0004         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
    0.1765   -0.0027   -0.2676    0.0317    0.0887   -0.0008    0.3040    0.0009   -0.0276   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0007   -0.0003    0.0004   -0.0013    0.0014   -0.0010    0.0009    0.0038   -0.0000    0.0005         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0072    0.0018    0.0074    0.0032   -0.0112    0.0090   -0.0276   -0.0000    0.0070   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0003   -0.0002    0.0004   -0.0003    0.0002   -0.0004   -0.0000    0.0005   -0.0000    0.0005         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0001   -0.0018    0.0000         0         0         0         0         0         0         0    0.0205         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
    0.9609   -0.0413    0.0330         0         0         0         0         0         0         0         0    0.3891    0.0220   -0.0175   -0.1197   -0.0076    0.0005    0.0001         0         0         0         0         0         0         0         0         0         0         0         0
    0.3663    0.2023    0.9184         0         0         0         0         0         0         0         0    0.0220    1.5669    0.0228   -0.4281   -0.0139   -0.0027    0.0002         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0521    0.0525    0.0043         0         0         0         0         0         0         0         0   -0.0175    0.0228    0.0229    0.0179    0.0111   -0.0004   -0.0001         0         0         0         0         0         0         0         0         0         0         0         0
   -0.3232    0.1088   -0.3224         0         0         0         0         0         0         0         0   -0.1197   -0.4281    0.0179    0.2497    0.0183    0.0024   -0.0004         0         0         0         0         0         0         0         0         0         0         0         0
      ⋮

show(atlas,q);

Input Arguments

robot — Robot model

rigidBodyTree object

Robot model, specified as a rigidBodyTree object. To use the massMatrix function, set the DataFormat property to either 'row' or 'column'.

configuration — Robot configuration

vector

Robot configuration, specified as a vector with positions for all nonfixed joints in the robot model. You can generate a configuration using homeConfiguration(robot), randomConfiguration(robot), or by specifying your own joint positions. To use the vector form of configuration, set the DataFormat property for the robot to either 'row' or 'column'.

Output Arguments

H — Mass matrix

positive-definite symmetric matrix

Mass matrix of the robot, returned as a positive-definite symmetric matrix with size n-by-n, where n is the velocity degrees of freedom of the robot.

More About

Dynamics Properties

When working with robot dynamics, specify the information for individual bodies of your manipulator robot using these properties of the rigidBody objects:

• Mass — Mass of the rigid body in kilograms.

• CenterOfMass — Center of mass position of the rigid body, specified as a vector of the form [x y z]. The vector describes the location of the center of mass of the rigid body, relative to the body frame, in meters. The centerOfMass object function uses these rigid body property values when computing the center of mass of a robot.

• Inertia — Inertia of the rigid body, specified as a vector of the form [Ixx Iyy Izz Iyz Ixz Ixy]. The vector is relative to the body frame in kilogram square meters. The inertia tensor is a positive definite matrix of the form:

  

  The first three elements of the Inertia vector are the moment of inertia, which are the diagonal elements of the inertia tensor. The last three elements are the product of inertia, which are the off-diagonal elements of the inertia tensor.

For information related to the entire manipulator robot model, specify these rigidBodyTree object properties:

• Gravity — Gravitational acceleration experienced by the robot, specified as an [x y z] vector in m/s². By default, there is no gravitational acceleration.

• DataFormat — The input and output data format for the kinematics and dynamics functions, specified as "struct", "row", or "column".

Dynamics Equations

Manipulator rigid body dynamics are governed by this equation:

$$\frac{d}{dt}\left[\begin{array}{c}q\\ \dot{q}\end{array}\right]=\left[\begin{array}{c}\dot{q}\\ M{(q)}^{-1}\left(-C(q,\dot{q})\dot{q}-G(q)-J{(q)}^T{f}_{Ext}+\tau \right)\end{array}\right]$$

also written as:

$$M(q)\ddot{q}=-C(q,\dot{q})\dot{q}-G(q)-J{(q)}^T{f}_{Ext}+\tau$$

where:

• $$M(q)$$ — is a joint-space mass matrix based on the current robot configuration. Calculate this matrix by using the massMatrix object function.

• $$C(q,\dot{q})$$ — are the Coriolis terms, which are multiplied by $$\dot{q}$$ to calculate the velocity product. Calculate the velocity product by using by the velocityProduct object function.

• $$G(q)$$ — is the gravity torques and forces required for all joints to maintain their positions in the specified gravity Gravity. Calculate the gravity torque by using the gravityTorque object function.

• $$J(q)$$ — is the geometric Jacobian for the specified joint configuration. Calculate the geometric Jacobian by using the geometricJacobian object function.

• $$f_{Ext}$$ — is a matrix of the external forces applied to the rigid body. Generate external forces by using the externalForce object function.

• $$\tau$$ — are the joint torques and forces applied directly as a vector to each joint.

• $$q,\dot{q},\ddot{q}$$ — are the joint configuration, joint velocities, and joint accelerations, respectively, as individual vectors. For revolute joints, specify values in radians, rad/s, and rad/s², respectively. For prismatic joints, specify in meters, m/s, and m/s².

To compute the dynamics directly, use the forwardDynamics object function. The function calculates the joint accelerations for the specified combinations of the above inputs.

To achieve a certain set of motions, use the inverseDynamics object function. The function calculates the joint torques required to achieve the specified configuration, velocities, accelerations, and external forces.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

When creating the rigidBodyTree object, use the syntax that specifies the MaxNumBodies as an upper bound for adding bodies to the robot model. You must also specify the DataFormat property as a name-value pair. For example:

robot = rigidBodyTree("MaxNumBodies",15,"DataFormat","row")

To minimize data usage, limit the upper bound to a number close to the expected number of bodies in the model. All data formats are supported for code generation. To use the dynamics functions, the data format must be set to "row" or "column".

The show and showdetails functions do not support code generation.

Version History

Introduced in R2017a

R2024a: Static memory allocation support

massMatrix now supports code generation with disabled dynamic memory allocation. For more information about disabling dynamic memory allocation, see Set Dynamic Memory Allocation Threshold (MATLAB Coder).