Compose external force matrix relative to base
Compute Forward Dynamics Due to External Forces on Rigid Body Tree Model
Calculate the resultant joint accelerations for a given robot configuration with applied external forces and forces due to gravity. A wrench is applied to a specific body with the gravity being specified for the whole robot.
Load a KUKA iiwa 14 robot model from the Robotics System Toolbox™
loadrobot. The robot is specified as a
Set the data format to "
row". For all dynamics calculations, the data format must be either "
row" or "
Set the gravity. By default, gravity is assumed to be zero.
kukaIIWA14 = loadrobot("kukaIiwa14",DataFormat="row",Gravity=[0 0 -9.81]);
Get the home configuration for the
q = homeConfiguration(kukaIIWA14);
Specify the wrench vector that represents the external forces experienced by the robot. Use the
externalForce function to generate the external force matrix. Specify the robot model, the end effector that experiences the wrench, the wrench vector, and the current robot configuration.
wrench is given relative to the "
iiwa_link_ee_kuka" body frame, which requires you to specify the robot configuration,
wrench = [0 0 0.5 0 0 0.3]; fext = externalForce(kukaIIWA14,"iiwa_link_ee_kuka",wrench,q);
Compute the resultant joint accelerations due to gravity, with the external force applied to the end-effector "
kukaIIWA14 is at its home configuration. The joint velocities and joint torques are assumed to be zero (input as an empty vector
qddot = forwardDynamics(kukaIIWA14,q,,,fext)
qddot = 1×7 -0.0023 -0.0112 0.0036 -0.0212 0.0067 -0.0075 499.9920
Compute Joint Torque to Counter External Forces
externalForce function to generate force matrices to apply to a rigid body tree model. The force matrix is an m-by-6 vector that has a row for each joint on the robot to apply a six-element wrench. Use the
externalForce function and specify the end effector to properly assign the wrench to the correct row of the matrix. You can add multiple force matrices together to apply multiple forces to one robot.
To calculate the joint torques that counter these external forces, use the
Load a Universal Robots UR5e from the Robotics System Toolbox™
loadrobot, specified as a
rigidBodyTree object. Update the gravity and set the data format to "
row". For all dynamics calculations, the data format must be either "
row" or "
manipulator = loadrobot("universalUR5e", DataFormat="row", Gravity=[0 0 -9.81]); showdetails(manipulator)
-------------------- Robot: (10 bodies) Idx Body Name Joint Name Joint Type Parent Name(Idx) Children Name(s) --- --------- ---------- ---------- ---------------- ---------------- 1 base base_link-base_fixed_joint fixed base_link(0) 2 base_link_inertia base_link-base_link_inertia fixed base_link(0) shoulder_link(3) 3 shoulder_link shoulder_pan_joint revolute base_link_inertia(2) upper_arm_link(4) 4 upper_arm_link shoulder_lift_joint revolute shoulder_link(3) forearm_link(5) 5 forearm_link elbow_joint revolute upper_arm_link(4) wrist_1_link(6) 6 wrist_1_link wrist_1_joint revolute forearm_link(5) wrist_2_link(7) 7 wrist_2_link wrist_2_joint revolute wrist_1_link(6) wrist_3_link(8) 8 wrist_3_link wrist_3_joint revolute wrist_2_link(7) flange(9) 9 flange wrist_3-flange fixed wrist_3_link(8) tool0(10) 10 tool0 flange-tool0 fixed flange(9) --------------------
Get the home configuration for
q = homeConfiguration(manipulator);
Set external force on
shoulder_link. The input wrench vector is expressed in the base frame.
fext1 = externalForce(manipulator,"shoulder_link",[0 0 0.0 0.1 0 0]);
Set external force on the end effector,
tool0. The input wrench vector is expressed in the
fext2 = externalForce(manipulator,"tool0",[0 0 0.0 0.1 0 0],q);
Compute the joint torques required to balance the external forces. To combine the forces, add the force matrices together. Joint velocities and accelerations are assumed to be zero (input as
tau = inverseDynamics(manipulator,q,,,fext1+fext2)
tau = 1×6 -0.0233 -52.4189 -14.4896 -0.0100 0.0100 -0.0000
robot — Robot model
Robot model, specified as a
rigidBodyTree object. To use the
externalForce function, set the
property to either
bodyname — Name of body to which external force is applied
string scalar | character vector
Name of body to which the external force is applied, specified as a string scalar or character
vector. This body name must match a body on the
wrench — Torques and forces applied to body
[Tx Ty Tz Fx Fy Fz] vector
Torques and forces applied to the body, specified as a
[Tx Ty Tz Fx Fy Fz]
vector. The first three elements of the wrench correspond to the moments around
xyz-axes. The last three elements are linear forces along the same
axes. Unless you specify the robot
configuration, the wrench is
assumed to be relative to the base frame.
configuration — Robot configuration
Robot configuration, specified as a vector with positions for all nonfixed joints in the robot
model. You can generate a configuration using
randomConfiguration(robot), or by specifying your own joint
positions. To use the vector form of
configuration, set the
DataFormat property for the
fext — External force matrix
n-by-6 matrix | 6-by-n matrix
External force matrix, returned as either an n-by-6 or
6-by-n matrix, where n is the velocity number
(degrees of freedom) of the robot. The shape depends on the
DataFormat property of
"row" data format uses an n-by-6 matrix. The
"column" data format uses a 6-by-n.
The composed matrix lists only values other than zero at the locations relevant to the body
specified. You can add force matrices together to specify multiple forces on multiple
bodies. Use the external force matrix to specify external forces to dynamics functions
When working with robot dynamics, specify the information for individual bodies of your manipulator robot using these properties of the
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
centerOfMassobject 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
Inertiavector 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:
Manipulator rigid body dynamics are governed by this equation:
also written as:
— is a joint-space mass matrix based on the current robot configuration. Calculate this matrix by using the
— are the Coriolis terms, which are multiplied by to calculate the velocity product. Calculate the velocity product by using by the
— is the geometric Jacobian for the specified joint configuration. Calculate the geometric Jacobian by using the
— is a matrix of the external forces applied to the rigid body. Generate external forces by using the
— are the joint torques and forces applied directly as a vector to each joint.
— 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.
 Featherstone, Roy. Rigid Body Dynamics Algorithms. Springer US, 2008. DOI.org (Crossref), doi:10.1007/978-1-4899-7560-7.
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
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
Introduced in R2017a