# Joint constraints and multibodies§

One of the most appealing features of a physics engine is to simulate articulations. Articulations, aka. joints, allow the restriction of the motion of one body part relative to another. For example, one well-known joint is the ball-in-socket joint also known as the ball joint: it allows one object to rotate freely with regard to the other but not to translate. This is typically used to simulate the shoulder of a ragdoll.

## Fundamental concepts§

Joints can be modeled in various ways but let’s talk about the concept of Degrees Of Freedom (DOF) first. In 3D, a rigid-body is capable of translating along the 3 coordinates axes $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$, and to rotate along those three axes as well. Therefore, a rigid-body is said to have 3 translational DOF and 3 rotational DOF. We can also say a 3D rigid-body has a total of 6 DOF. The 2D case is similar but with less possibilities of movements: a 2D rigid-body has 2 translational DOF and only 1 rotational DOF (which forms a total of 3 DOF). The number of relative DOF of a body part wrt. another body part is the number of possible relative translations and rotations.

Note

The Ground body on the other hand cannot move at all. Therefore it has 0 DOFs. Moreover a multibody has a number of DOF equal to the sum of the number of DOF of its multibody links.

Warning

The number of DOF of a body can be retrieved by body.ndofs(). The status of a body (dynamic, static, kinematic) does not affect the result of body.ndofs() even if strictly speaking a static or kinematic body should be seen as having no DOF (because it cannot move freely wrt. any axis). The body.status_dependent_ndofs() method will take this body status into account and return 0 for static and kinematic bodies.

The goal of a joint is to reduce the number of DOF a body part has. For example, the aforementioned ball joint removes all relative translations between two body parts. Therefore, it allows only the 3 rotational DOF in 3D simulations or the 1 rotational DOF in 2D simulations. Other joints exist allowing other combinations of relative DOF. Note that because there are less possible motions in 2D, some joints are only defined in 3D. This is illustrated by empty cells in the following table for joints that are not defined in 2D:

Joint Allowed DOF in 2D Allowed DOF in 3D
Fixed joint None None
Free joint All All
Prismatic joint 1 Translation 1 Translation
Revolute joint 1 Rotation 1 Rotation
Ball joint 1 Rotation 3 Rotations
Cartesian joint 2 Translations 3 Translations
Planar joint 2 Translations + 1 Rotation
Cylindrical joint 1 Translation + 1 Rotation (along the same axis)
Pin-slot joint 1 Translation + 1 Rotation (along different axes)
Rectangular joint 2 Translations
Universal joint 2 Rotations

In 3D, a special Helical joint also exists: it allows only one DOF which is a bit special as it is the combination of a rotation and a translation. In other words, a body part attached to the ground by an helical joint will only be able to translate and rotate simultaneously: any translation induce automatically a rotation and vice-versa.

In practice, there are two main ways of modeling joints. Both are implemented by nphysics because each have very different advantages and limitations:

1. The reduced-coordinates approach encodes the reduction of DOF directly into the equations of motion. For example, a 3D rigid-body attached to the ground with a revolute joint will have its position encoded by only one variable: the rotation angle. Therefore, integrating its motion only changes this one variable and doesn’t need additional forces or mathematical constraints to be generated. The clear advantage is that there is no way for the physics engine to apply any motion other than that single rotation to this body, meaning there is no way the body shifts to a position that is not realistic, even if the dynamics solver does not converge completely.
2. The constraints-based approach (or full-coordinates approach) is the most commonly available approach on other physics engines for video-games and animations. Here, a 3D rigid-body attached to the ground with a revolute joint will still have its position encoded by 6 variables (3 for translations and 3 for rotations) just like any rigid-body without a joint. Then the integrator will add mathematical constraints to the dynamic system to ensure forces are applied to simulate the reduction of the number of DOF as imposed by the joints. In practice, this means that the rigid-body will break the joint constraint if the constraint solver does not converge completely.

This description shows only one aspect of the difference between the reduced-coordinates approach and the constraints-based approach. More generally, the reduced-coordinates approach favors accuracy while the constraints-based approach favors versatility. The following table compares the advantages and limitations of both approaches:

Reduced-coordinates approach Constraints-based approach
Joints cannot be violated at all. Joints can be violated if the solver does not converge.
Moderately large time-steps are possible. Moderately large time-steps may make the simulation explode.
Large assemblies are stable. Large assemblies easily break without a large number of solver iterations.
Joint forces are never computed explicitly, thus cannot be retrieved. Joint forces are always computed and can be retrieved.
Topological restriction: body parts must be linked following a tree structure. The link between body parts can form any graph.

The following schematics illustrate a configuration that can be simulated by a multibody (left assembly with a tree structure), and one that cannot (right assembly with a graph structure). The assembly on the left models a SCARA robotic arm with 3 rotational DOF (due to three revolute joints) and 1 translational DOF (due to one prismatic joint). The assembly on the right models a necklace with five pearls. It has a total of 15 rotational DOF (due to five ball joints):

Which approach should I use?

The choice of approach depends on the application. For robotics, the reduced-coordinates approach is generally preferred because of its accuracy and ease of use, e.g., for control, inverse kinematics, etc.

Video games traditionally favor the constraints-based approach since most existing physics libraries implement only this. Moreover if joint assemblies are small, and joints are frequently added and removed, the constraints-based approach will be more efficient. Some other physics libraries implement the reduced-coordinates approach as well but often using the Featherstone algorithm which is extremely unstable in practice.

Simulating closed loops like for a necklace cannot be achieved with the reduced-coordinates approach only. However, it is possible to combine both approaches by using joint constraints only to close the loops. Refer to the last section for details.

The use of the reduced-coordinates approach is detailed in the multibodies section and demonstrated by the Multibody joints demo. The constraints-based approach is detailed in the joint constraints section and demonstrated by the Joint constraints demo.

## Multibodies§

Multibodies implement the reduced-coordinates approach. A multibody is a set of multibody links attached together by a multibody joint.

### Creating a multibody§

Creating a multibody is done link-by-link using the MultibodyDesc structure based on the builder pattern. Each link of a multibody is described by a single MultibodyDesc to which children links can be added by calling the .add_child method. This method will itself return a new MultibodyDesc that can be used to initialize the child, create even more nested children, etc.

use na::{Vector2, Point2, Isometry2, Matrix2};
use nphysics2d::object::MultibodyDesc;
use nphysics2d::joint::{RevoluteJoint, PrismaticJoint};
use nphysics2d::math::{Velocity, Inertia};

let joint = RevoluteJoint::new(-0.1);
let mut multibody_desc = MultibodyDesc::new(joint)
// The velocity of this body.
// Default: zero velocity.
.velocity(Velocity::linear(1.0, 2.0))
// The angular inertia tensor of this rigid body, expressed on its local-space.
// Default: the zero matrix.
.angular_inertia(3.0)
// The rigid body mass.
// Default: 0.0
.mass(1.2)
// The mass and angular inertia of this rigid body expressed in
// its local-space. Default: zero.
// Will override .mass(...) and .angular_inertia(...).
.local_inertia(Inertia::new(1.0, 3.0))
// The center of mass of this rigid body expressed in its local-space.
// Default: the origin.
.local_center_of_mass(Point2::new(1.0, 2.0))
/// The position of the joint wrt. parent, expressed in the local
/// frame of parent.
/// Default: Vector2::zeros()
.parent_shift(Vector2::new(1.0, 2.0))
/// The position of the newly created multibody link wrt. the joint,
/// expressed in the local frame of the joint.
/// Default: Vector2::zeros()
.body_shift(Vector2::new(1.0, 2.0));

/// Add a children link to the multibody link represented by multibody_desc.
let child_joint = PrismaticJoint::new(Vector2::y_axis(), 0.0);
// The add_child method returns another MultibodDesc used to
// set the properties of the child. It is also possible to call
// .add_child on this child.
.body_shift(Vector2::x() * 2.0);

/// We can add a second child to our multibody by re-using the initial multibody_desc:
let second_child_joint = RevoluteJoint::new(2.3);

/// Actually build the multibody.
let multibody = multibody_desc.build();
// Finally, wa may add this multibody to a body set.
let multibody_handle = body_set.insert(multibody);

use na::{Vector3, Point3, Isometry3, Matrix3};
use nphysics3d::object::MultibodyDesc;
use nphysics3d::joint::{RevoluteJoint, PrismaticJoint, HelicalJoint};
use nphysics3d::math::{Velocity, Inertia};

let joint = RevoluteJoint::new(-0.1);
let mut multibody_desc = MultibodyDesc::new(joint)
// The name of this multibody link.
// Default: ""
// The velocity of this body.
// Default: zero velocity.
.velocity(Velocity::linear(1.0, 2.0, 3.0))
// The angular inertia tensor of this rigid body, expressed on its local-space.
// Default: the zero matrix.
.angular_inertia(Matrix3::from_diagonal_element(3.0))
// The rigid body mass.
// Default: 0.0
.mass(1.2)
// The mass and angular inertia of this rigid body expressed in
// its local-space. Default: zero.
// Will override .mass(...) and .angular_inertia(...).
.local_inertia(Inertia::new(1.0, Matrix3::from_diagonal_element(3.0)))
// The center of mass of this rigid body expressed in its local-space.
// Default: the origin.
.local_center_of_mass(Point3::new(1.0, 2.0, 3.0))
/// The position of the joint wrt. parent, expressed in the local
/// frame of parent.
/// Default: Vector3::zeros()
.parent_shift(Vector3::new(1.0, 2.0, 3.0))
/// The position of the newly created multibody link wrt. the joint,
/// expressed in the local frame of the joint.
/// Default: Vector3::zeros()
.body_shift(Vector3::new(1.0, 2.0, 3.0));

/// Add a children link to the multibody link represented by multibody_desc.
let child_joint = PrismaticJoint::new(Vector2::y_axis(), 0.0);
// The add_child method returns another MultibodDesc used to
// set the properties of the child. It is also possible to call
// .add_child on this child.
.body_shift(Vector3::x() * 2.0);

/// We can add a second child to our multibody by re-using the initial multibody_desc:
let second_child_joint = HelicalJoint::new(Vector3::y_axis(), 1.0, 0.0);

/// Actually build the multibody.
let multibody = multibody_desc.build();
// Finally, wa may add this multibody to a body set.
let multibody_handle = body_set.insert(multibody);

The following table summarizes the types corresponding to the joints mentioned at the beginning of this chapter that can be used for the MultibodyDesc:

Joint name Multibody joint type on nphysics
Fixed joint FixedJoint
Prismatic joint PrismaticJoint
Revolute joint RevoluteJoint
Ball joint BallJoint
Cartesian joint CartesianJoint
Planar joint PlanarJoint
Cylindrical joint CylindricalJoint
Pin-slot joint PinSlotJoint
Rectangular joint RectangularJoint
Universal joint UniversalJoint

Note

The first multibody link of a multibody is necessarily attached to an implicit static ground. Note however that “attached” is a bit misleading here. Indeed if joint is set to an instance of FreeJoint, then this first multibody link will have all the possible degrees of freedom, making it completely free to perform any movement.

Warning

The FreeJoint can be used only by the first link of the multibody otherwise, the creation of the multibody will panic.

While it is not possible to remove a link from an existing multibody, it is possible to add new links:

// This will create the multibody links identified by multibody_desc
// and all its children. The multibody link identified by multibody_desc
// is then attached as a child of the pre-existing multibody link identified
// by parent_handle.
multibody_desc.build_with_parent(multibody, link_id);

The BodyPartHandle identifying a specific link of a multibody can be constructed with BodyPartHandle(handle, i) where handle is the multibody handle, and i designates the i-th link of the multibody. Multibody links are indexed in their creation order.

Alternatively, you may retrieve a reference to a multibody link and its index using its name set during its construction:

for link in multibody.links_with_name("my multibody link name") {
/// ...
}

You may refer to the code of that demo for concrete examples of multibody creation.

### Removing a multibody§

The removal of a multibody from the body set uses the same method as the removal of a rigid-body: body_set.remove(handle). It is not possible to remove a single link of a multibody without removing the whole multibody altogether.

### Multibody joint limits and motors§

It is often desirable to limit the amplitude of movement a multibody link can have with regard to its parent. For example we might want to limit the minimum and maximum value for the DOF of a prismatic joint in order to simulate a piston with finite stroke. Or we might want to limit the maximum angle a revolute joint can make with regard to its parent. Those are modeled by joint limits. Joint limits are currently only implemented for multibody joints with DOF that are independent from each other. Therefore, it is not implemented for the FreeJoint, BallJoint, and CartesianJoint. All other joints have methods similar to the following:

Method Description
.enable_min_angle(limit) Sets the minimum angle (in radians) the joint angular DOF can take.
.enable_max_angle(limit) Sets the maximum angle (in radians) the joint angular DOF can take.
.disable_min_angle() Disable the lower angle limit.
.disable_max_angle() Disable the upper angle limit.
.enable_min_offset(limit) Sets the minimum offset the joint linear DOF can take.
.enable_max_offset(limit) Sets the maximum offset the joint linear DOF can take.
.disable_min_offset() Disable the lower offset limit.
.disable_max_offset() Disable the upper offset limit.

Note

Joints with no angular DOF will not have the methods related to the angular limits. Similarly, joints with no linear DOF will not have the methods related to the linear limits. Joints with several angular or linear DOF will have those methods with an index appended to their name, e.g., the .enable_min_angle_1(limit) method of an universal joint will enable a lower limit for its first angular DOF.

It is also often desirable to motorize a joint to impose a movement. For example simulating a motorized car wheel can be achieved by enabling a motor on a revolute joint linking the wheel with the car frame. A motor is specified by a target velocity and a maximum force. nphysics will apply forces at the motorized joint so that the joint reaches the target velocity, but will never apply a force that is stronger than some maximum user-defined value. Setting a small maximum force can be useful for having the joint accelerate progressively. In any case, setting a maximum motor force is highly recommended.

Warning

Not setting any maximum force implies that nphysics is allowed to generate any force as large as necessary to reach the target velocity in one timestep. This is strongly discouraged because that can lead to odd behaviors. For example if a motorized pendulum with no maximum force hits a wall, the non-penetration constraint due to the contact will have to generate a force that counteracts the effect of the motor force. However this will not be possible given the motor is free to apply any, potentially infinite, force. This can result in the pendulum traversing the wall.

For the moment, joint motors are only implemented for multibody joints with DOF that are independent from each other. Therefore, it is not implemented for FreeJoint, BallJoints, and CartesianJoints. All other joints have methods similar to the following:

Method Description
.enable_angular_motor() Activate the angular motor.
.disable_angular_motor() Disable the angular motor.
.set_desired_angular_motor_velocity(vel) Set the angular velocity the motor must achieve. Setting this to 0.0 is possible and will simulate joint friction.
.set_max_angular_motor_torque(max) Sets the maximum torque the motor can deliver to reach the desired velocity.
.enable_linear_motor() Activate the linear motor.
.disable_linear_motor() Disable the linear motor.
.set_desired_linear_motor_velocity(vel) Set the velocity the motor must achieve. Setting this to 0.0 is possible and will simulate joint friction.
.set_max_linear_motor_force(max) Sets the maximum linear force the motor can deliver to reach the desired velocity.

Note

Joints with no angular DOF will not have the methods related to the angular motors. Similarly, joints with no linear DOF will not have the methods related to the linear motors. Joints with several angular or linear DOF will have those methods with an index appended to their name, e.g., the .enable_angular_motor_1() method of an universal joint will enable a motor for its first angular DOF.

## Joint constraints§

Joint constraints implement the constraints-based approach. The following table summarizes the types corresponding to the joints mentioned at the beginning of this chapter:

Joint name Joint constraint type on nphysics
Fixed joint FixedConstraint
Prismatic joint PrismaticConstraint
Revolute joint RevoluteConstraint
Ball joint BallConstraint
Cartesian joint CartesianConstraint
Planar joint PlanarConstraint
Cylindrical joint CylindricalConstraint
Pin-slot joint PinSlotConstraint
Rectangular joint RectangularConstraint
Universal joint UniversalConstraint

A joint constraint is completely configured at its creation, and added to the constraint set by the constraint_set.insert(constraint) method. Each joint constraint requires specific information for being constructed, but all roughly need:

1. The handles of the two body parts attached at each end of the joint. Handle of any type of body part is accepted. This includes rigid-bodies, Ground bodies, an element of a deformable body, as well as a multibody link. Attaching a joint to a multibody link can be especially useful to handle complex assemblies with loops as described in the next section.
2. The position of the joint endpoints with regard to each body part. A joint endpoint is often referred to as an anchor throughout the documentation of nphysics.

You may refer to the code of this demo for concrete examples of joint constraint configurations.

## Combining both§

Combining multibodies and joint constraints is a useful way of combining the stability of multibodies with the flexibility of joint constraints. Indeed, one of the most appealing features of a multibody is its stability and ease of use (especially for robotics). However its greatest weakness is its inability to represent assemblies that do not match a tree structure, i.e., an articulated body composed of graph-like assembly of solids (each graph node being a solid and each graph edge being an articulation) cannot be simulated by a multibody. A common approach is thus to:

1. Define a multibody from a spanning-tree of the graph.
2. Create joint constraints for each articulation missing from this multibody to complete the graph. Those joint constraints are therefore attached to two multibody links. They are often called “loop-closing constraints” since they close the loops of the assembly’s graph structure.

The following shows an example of combination of multibodies and joint constraints for the simulation of a necklace. It is composed of 5 pearls forming a single loop attached together by 5 ball joints. Since such a loop cannot be simulated by a multibody, we first start to create 5 multibody links attached together with 4 BallJoint. Only 4 joints can be added here since a 5th would close the loop. The 5th joint that closes the loop must be modeled as a joint constraint, here a BallConstraint between the first and the last link:

Note

Note that using the world to create multibody links removes any risk of inadvertently creating multibody links attached in such a way that they would form anything but a tree structure.