## Robot Dynamics

This topic details the different elements, properties, and equations of rigid body robot
dynamics. *Robot dynamics* are the relationship between the forces
acting on a robot and the resulting motion of the robot. In Robotics System Toolbox™, manipulator dynamics information is contained within a `rigidBodyTree`

object, which specifies the rigid bodies, attachment points, and
inertial parameters for both kinematics and dynamics calculations.

**Note**

To use dynamics object functions, you must set the `DataFormat`

property of the
`rigidBodyTree`

object to
`"row"`

or `"column"`

. These setting accept inputs
and return outputs as row or column vectors, respectively, for relevant robotics
calculations, such as robot configurations or joint torques.

### 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/s^{2}. 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/s

^{2}, respectively. For prismatic joints, specify in meters, m/s, and m/s^{2}.

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.

## See Also

### Functions

`forwardDynamics`

|`inverseDynamics`

|`externalForce`

|`geometricJacobian`

|`gravityTorque`

|`centerOfMass`

|`massMatrix`

|`velocityProduct`