# 6. Symbolics using SymPy

This example is also available as a Jupyter notebook that can be run locally The notebook can be found in the examples directory of the package. If the notebooks are missing, you may need to run using Pkg; Pkg.build().

## Setup

using Pkg # hide
Pkg.activate("../../../../examples/6. Symbolics using SymPy") # hide
Pkg.instantiate() # hide
using RigidBodyDynamics
using StaticArrays
using SymPy

## Create symbolic parameters

• Masses: $m_1, m_2$
• Mass moments of inertia (about center of mass): $I_1, I_2$
• Link lengths: $l_1, l_2$
• Center of mass locations (w.r.t. preceding joint axis): $c_1, c_2$
• Gravitational acceleration: $g$
inertias = @syms m_1 m_2 I_1 I_2 positive = true
lengths = @syms l_1 l_2 c_1 c_2 real = true
gravitational_acceleration = @syms g real = true
params = [inertias..., lengths..., gravitational_acceleration...]
transpose(params)

## Create double pendulum Mechanism

A Mechanism contains the joint layout and inertia parameters, but no state information.

T = Sym # the 'scalar type' of the Mechanism we'll construct
axis = SVector(zero(T), one(T), zero(T)) # axis of rotation for each of the joints
double_pendulum = Mechanism(RigidBody{T}("world"); gravity = SVector(zero(T), zero(T), g))
world = root_body(double_pendulum) # the fixed 'world' rigid body

Attach the first (upper) link to the world via a revolute joint named 'shoulder'

inertia1 = SpatialInertia(CartesianFrame3D("upper_link"),
moment=I_1 * axis * transpose(axis),
com=SVector(zero(T), zero(T), c_1),
mass=m_1)
body1 = RigidBody(inertia1)
joint1 = Joint("shoulder", Revolute(axis))
joint1_to_world = one(Transform3D{T}, frame_before(joint1), default_frame(world));
attach!(double_pendulum, world, body1, joint1,
joint_pose = joint1_to_world);
nothing #hide

Attach the second (lower) link to the world via a revolute joint named 'elbow'

inertia2 = SpatialInertia(CartesianFrame3D("lower_link"),
moment=I_2 * axis * transpose(axis),
com=SVector(zero(T), zero(T), c_2),
mass=m_2)
body2 = RigidBody(inertia2)
joint2 = Joint("elbow", Revolute(axis))
joint2_to_body1 = Transform3D(
frame_before(joint2), default_frame(body1), SVector(zero(T), zero(T), l_1))
attach!(double_pendulum, body1, body2, joint2,
joint_pose = joint2_to_body1)

## Create MechanismState associated with the double pendulum Mechanism

A MechanismState stores all state-dependent information associated with a Mechanism.

x = MechanismState(double_pendulum);
nothing #hide

Set the joint configuration vector of the MechanismState to a new vector of symbolic variables

q = configuration(x)
for i in eachindex(q)
q[i] = symbols("q_$i", real = true) end Set the joint velocity vector of the MechanismState to a new vector of symbolic variables v = velocity(x) for i in eachindex(v) v[i] = symbols("v_$i", real = true)
end

## Compute dynamical quantities in symbolic form

Mass matrix

simplify.(mass_matrix(x))

Kinetic energy

simplify(kinetic_energy(x))

Potential energy

simplify(gravitational_potential_energy(x))