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Merge pull request #146 from tkoolen/sympy
Add symbolic dynamics example using SymPy.jl
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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Setup" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 2, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [], | ||
| "source": [ | ||
| "using RigidBodyDynamics\n", | ||
| "using StaticArrays\n", | ||
| "using SymPy" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Create symbolic parameters\n", | ||
| "* Masses: $m_1, m_2$\n", | ||
| "* Mass moments of inertia (about center of mass): $I_1, I_2$\n", | ||
| "* Link lengths: $l_1, l_2$\n", | ||
| "* Center of mass locations (w.r.t. preceding joint axis): $c_1, c_2$\n", | ||
| "* Gravitational acceleration: $g$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 3, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/latex": [ | ||
| "\\begin{bmatrix}m_{1}\\\\m_{2}\\\\I_{1}\\\\I_{2}\\\\l_{1}\\\\l_{2}\\\\c_{1}\\\\c_{2}\\\\g\\end{bmatrix}" | ||
| ], | ||
| "text/plain": [ | ||
| "9-element Array{SymPy.Sym,1}\n", | ||
| "[m_1]\n", | ||
| "[ ]\n", | ||
| "[m_2]\n", | ||
| "[ ]\n", | ||
| "[I_1]\n", | ||
| "[ ]\n", | ||
| "[I_2]\n", | ||
| "[ ]\n", | ||
| "[l_1]\n", | ||
| "[ ]\n", | ||
| "[l_2]\n", | ||
| "[ ]\n", | ||
| "[c_1]\n", | ||
| "[ ]\n", | ||
| "[c_2]\n", | ||
| "[ ]\n", | ||
| "[ g ]" | ||
| ] | ||
| }, | ||
| "execution_count": 3, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "inertias = @syms m_1 m_2 I_1 I_2 positive = true\n", | ||
| "lengths = @syms l_1 l_2 c_1 c_2 real = true\n", | ||
| "gravitationalAcceleration = @syms g real = true\n", | ||
| "params = [inertias..., lengths..., gravitationalAcceleration...]" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Create double pendulum `Mechanism`\n", | ||
| "\n", | ||
| "A `Mechanism` contains the joint layout and inertia parameters, but no state information." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 4, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/plain": [ | ||
| "Vertex: world (root)\n", | ||
| " Vertex: upper_link, Edge: shoulder\n", | ||
| " Vertex: lower_link, Edge: elbow" | ||
| ] | ||
| }, | ||
| "execution_count": 4, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "T = Sym # the 'scalar type' of the Mechanism we'll construct\n", | ||
| "axis = SVector(zero(T), one(T), zero(T)) # axis of rotation for each of the joints\n", | ||
| "doublePendulum = Mechanism(RigidBody{T}(\"world\"); gravity = SVector(0, 0, g))\n", | ||
| "world = root_body(doublePendulum) # the fixed 'world' rigid body\n", | ||
| "\n", | ||
| "# Attach the first (upper) link to the world via a revolute joint named 'shoulder'\n", | ||
| "inertia1 = SpatialInertia(CartesianFrame3D(\"upper_link\"), I_1 * axis * axis', SVector(0, 0, c_1), m_1)\n", | ||
| "body1 = RigidBody(inertia1)\n", | ||
| "joint1 = Joint(\"shoulder\", Revolute(axis))\n", | ||
| "joint1ToWorld = Transform3D{T}(joint1.frameBefore, default_frame(world))\n", | ||
| "attach!(doublePendulum, world, joint1, joint1ToWorld, body1)\n", | ||
| "\n", | ||
| "# Attach the second (lower) link to the world via a revolute joint named 'elbow'\n", | ||
| "inertia2 = SpatialInertia(CartesianFrame3D(\"lower_link\"), I_2 * axis * axis', SVector(0, 0, c_2), m_2)\n", | ||
| "body2 = RigidBody(inertia2)\n", | ||
| "joint2 = Joint(\"elbow\", Revolute(axis))\n", | ||
| "joint2ToBody1 = Transform3D(joint2.frameBefore, default_frame(body1), SVector(0, 0, l_1))\n", | ||
| "attach!(doublePendulum, body1, joint2, joint2ToBody1, body2)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Create `MechanismState` associated with the double pendulum `Mechanism`\n", | ||
| "\n", | ||
| "A `MechanismState` stores all state-dependent information associated with a `Mechanism`." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 5, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [], | ||
| "source": [ | ||
| "x = MechanismState(T, doublePendulum);" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 6, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/latex": [ | ||
| "\\begin{bmatrix}q_{1}\\\\q_{2}\\end{bmatrix}" | ||
| ], | ||
| "text/plain": [ | ||
| "2-element Array{SymPy.Sym,1}\n", | ||
| "[q_1]\n", | ||
| "[ ]\n", | ||
| "[q_2]" | ||
| ] | ||
| }, | ||
| "execution_count": 6, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "# Set the joint configuration vector of the MechanismState to a new vector of symbolic variables\n", | ||
| "q = configuration_vector(x)[:] = [symbols(\"q_$i\", real = true) for i = 1 : num_positions(x)]" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 7, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/latex": [ | ||
| "\\begin{bmatrix}v_{1}\\\\v_{2}\\end{bmatrix}" | ||
| ], | ||
| "text/plain": [ | ||
| "2-element Array{SymPy.Sym,1}\n", | ||
| "[v_1]\n", | ||
| "[ ]\n", | ||
| "[v_2]" | ||
| ] | ||
| }, | ||
| "execution_count": 7, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "# Set the joint velocity vector of the MechanismState to a new vector of symbolic variables\n", | ||
| "v = velocity_vector(x)[:] = [symbols(\"v_$i\", real = true) for i = 1 : num_positions(x)]" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Compute dynamical quantities in symbolic form" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 8, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/latex": [ | ||
| "\\begin{bmatrix}I_{1} + I_{2} + 2 c_{2} l_{1} \\cos{\\left (q_{2} \\right )} + l_{1}^{2} m_{2}&I_{2} + c_{2} l_{1} \\cos{\\left (q_{2} \\right )}\\\\I_{2} + c_{2} l_{1} \\cos{\\left (q_{2} \\right )}&I_{2}\\end{bmatrix}" | ||
| ], | ||
| "text/plain": [ | ||
| "2×2 Array{SymPy.Sym,2}\n", | ||
| "[ 2 ]\n", | ||
| "[I_1 + I_2 + 2*c_2*l_1*cos(q_2) + l_1 *m_2 I_2 + c_2*l_1*cos(q_2)]\n", | ||
| "[ ]\n", | ||
| "[ I_2 + c_2*l_1*cos(q_2) I_2 ]" | ||
| ] | ||
| }, | ||
| "execution_count": 8, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "# Mass matrix\n", | ||
| "M = mass_matrix(x)\n", | ||
| "function simplify!(a::Array{SymPy.Sym})\n", | ||
| " for i in eachindex(a)\n", | ||
| " a[i] = simplify(a[i])\n", | ||
| " end\n", | ||
| " a\n", | ||
| "end\n", | ||
| "simplify!(M.data) # Note: M is a Symmetric matrix type; need to simplify the underlying data\n", | ||
| "full(M) # convert to full form so that it is pretty-printed (minor bug in SymPy.jl)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 9, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/latex": [ | ||
| "$$\\frac{I_{1} v_{1}^{2}}{2} + \\frac{I_{2} v_{1}^{2}}{2} + I_{2} v_{1} v_{2} + \\frac{I_{2} v_{2}^{2}}{2} + c_{2} l_{1} v_{1}^{2} \\cos{\\left (q_{2} \\right )} + c_{2} l_{1} v_{1} v_{2} \\cos{\\left (q_{2} \\right )} + \\frac{l_{1}^{2} m_{2}}{2} v_{1}^{2}$$" | ||
| ], | ||
| "text/plain": [ | ||
| " 2 2 2 \n", | ||
| "I_1*v_1 I_2*v_1 I_2*v_2 2 \n", | ||
| "-------- + -------- + I_2*v_1*v_2 + -------- + c_2*l_1*v_1 *cos(q_2) + c_2*l_1\n", | ||
| " 2 2 2 \n", | ||
| "\n", | ||
| " 2 2\n", | ||
| " l_1 *m_2*v_1 \n", | ||
| "*v_1*v_2*cos(q_2) + -------------\n", | ||
| " 2 " | ||
| ] | ||
| }, | ||
| "execution_count": 9, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "# Kinetic energy\n", | ||
| "simplify(kinetic_energy(x))" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 10, | ||
| "metadata": { | ||
| "collapsed": false | ||
| }, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/latex": [ | ||
| "$$- g \\left(c_{1} \\cos{\\left (q_{1} \\right )} + c_{2} \\cos{\\left (q_{1} + q_{2} \\right )} + l_{1} m_{2} \\cos{\\left (q_{1} \\right )}\\right)$$" | ||
| ], | ||
| "text/plain": [ | ||
| "-g*(c_1*cos(q_1) + c_2*cos(q_1 + q_2) + l_1*m_2*cos(q_1))" | ||
| ] | ||
| }, | ||
| "execution_count": 10, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "# Potential energy\n", | ||
| "simplify(potential_energy(x))" | ||
| ] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Julia 0.5.0", | ||
| "language": "julia", | ||
| "name": "julia-0.5" | ||
| }, | ||
| "language_info": { | ||
| "file_extension": ".jl", | ||
| "mimetype": "application/julia", | ||
| "name": "julia", | ||
| "version": "0.5.0" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 1 | ||
| } |
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