# Conversion to a system of second order ODEs then first order ODEs

Posted 2 months ago
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 Dear Colleagues,I have used Andrew Moylan's most excellent blog on the n-link pendulum on a cart as inspiration for my own notebook to derive the Equations of Motion (EoMs) for the pendulum. I have included viscous and friction in the revolute joints and my notation is slightly different than Andrew's but the structure of the EoMs is similar. For example, for a two-link pendulum I get. {Subscript[b, 0] Derivative[1][Subscript[q, 0]][t] + Subscript[m, 0] (Subscript[q, 0]^\[Prime]\[Prime])[t] + Subscript[m, 1] (Subscript[q, 0]^\[Prime]\[Prime])[t] + Subscript[m, 2] (Subscript[q, 0]^\[Prime]\[Prime])[t] + 1/2 Cos[Subscript[q, 2][t]] Subscript[l, 2] Subscript[m, 2] (Subscript[q, 2]^\[Prime]\[Prime])[t] == u[t] + 1/2 (Sin[Subscript[q, 2][t]] Subscript[l, 2] Subscript[m, 2] Derivative[1][Subscript[q, 2]][t]^2 + Subscript[l, 1] (Subscript[m, 1] + 2 Subscript[m, 2]) (Sin[Subscript[q, 1][t]] Derivative[1][Subscript[q, 1]][ t]^2 - Cos[Subscript[q, 1][t]] (Subscript[q, 1]^\[Prime]\[Prime])[t])), \!$$\*SubsuperscriptBox[\(l$$, $$1$$, $$2$$]\ $$( \*SubscriptBox[\(m$$, $$1$$] + 4\ \*SubscriptBox[$$m$$, $$2$$])\)\ $$\* SuperscriptBox[ SubscriptBox["q", "1"], "\[Prime]\[Prime]", MultilineFunction->None][t]$$\) + 4 (Subscript[k, 1] Subscript[q, 1][t] + Subscript[k, 2] (Subscript[q, 1][t] - Subscript[q, 2][t]) + Subscript[b, 1] Derivative[1][Subscript[q, 1]][t] + Subscript[moi, 1] (Subscript[q, 1]^\[Prime]\[Prime])[t]) + 2 Subscript[l, 1] (Subscript[m, 1] (-g Sin[Subscript[q, 1][t]] + Cos[Subscript[q, 1][t]] (Subscript[q, 0]^\[Prime]\[Prime])[ t]) + Subscript[m, 2] (-2 g Sin[Subscript[q, 1][t]] + 2 Cos[Subscript[q, 1][t]] (Subscript[q, 0]^\[Prime]\[Prime])[ t] + Subscript[l, 2] (Sin[Subscript[q, 1][t] - Subscript[q, 2][t]] Derivative[ 1][Subscript[q, 2]][t]^2 + Cos[Subscript[q, 1][t] - Subscript[q, 2][t]] (Subscript[q, 2]^\[Prime]\[Prime])[t]))) == 0, Subscript[k, 2] (-Subscript[q, 1][t] + Subscript[q, 2][t]) + Subscript[b, 2] Derivative[1][Subscript[q, 2]][t] + 1/4 (\!$$\*SubsuperscriptBox[\(l$$, $$2$$, $$2$$]\ \*SubscriptBox[$$m$$, $$2$$]\) + 4 Subscript[moi, 2]) (Subscript[q, 2]^\[Prime]\[Prime])[t] == 1/2 Subscript[l, 2] Subscript[m, 2] (g Sin[Subscript[q, 2][t]] - Cos[Subscript[q, 2][t]] (Subscript[q, 0]^\[Prime]\[Prime])[t] + Subscript[l, 1] (Sin[Subscript[q, 1][t] - Subscript[q, 2][t]] Derivative[1][ Subscript[q, 1]][t]^2 - Cos[Subscript[q, 1][t] - Subscript[q, 2][t]] (Subscript[q, 1]^\[Prime]\[Prime])[t]))} where q0 is the position of the cart and q1 and q2 are the angles of the links measured from the positive vertical. I have a requirement to cast the EoMs in a form that is called, in some texts, State-Dependent Coefficient form. Right now I am doing this by hand. In the first step I convert the EoMs above to a system of coupled second order ODEs of the form$M(\mathbf{q}) \ddot{\mathbf{q}} + C(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}} + K(\mathbf{q}) \mathbf{q} = 0.$In the second step I use standard matrix manipulation to get to the state space form of the EoMs $\dot{\mathbf{x}} = A(\mathbf{x}, t) \mathbf{x},$where the state vector is $\mathbf{x}=(\mathbf{q},\dot{\mathbf{q}})^\text{T}$While this is time consuming but manageable for a two-link pendulum it becomes quite tedious and prone to errors for three or more links.I have tried to find a way to program the conversion from the EoMs generated with Mathematica to the second order form using Simplify[] with assumptions but I have made no progress. Are there any documented methods for symbolic manipulation to convert the EoMs to a vector equation? I'm sure folks in robotics have figured this out already.Thank you!Bogdan
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Posted 2 months ago
 Neil,Thank you for your suggestions. I should have opened my question with details of the application.I am working on a few control and estimation algorithms.The system is highly dynamical and I have to operate away from equilibrium so linearization is a last resort. One of the nonlinear controls uses the solution of the State Dependent Riccati Equation (SDRE) that requires the dynamics of the system to be cast in the State-Dependent Coefficients (SDC) form. $\dot{\mathbf{x}}=A(\mathbf{x}) \mathbf{x}+B(\mathbf{x}) \mathbf{u}$ I have to verify all sort of requirements on the $A(\mathbf{x})$ and $B(\mathbf{x})$ matrices and I would like to do that by hand or symbolically. Moreover, the control designer has quite a bit of freedom in converting the nonlinear system state function to the SDC form, i.e., $f(\mathbf{x} = A(\mathbf{x}) \mathbf{x}$. Some papers suggest that this factorization can be somewhat optimized and I want to put that to test.For the estimator I would like to have a "truth model" of the dynamics so I can determine on-line the "goodness" of the estimation.Bogdan
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Posted 2 months ago
 Bogdan,I have two suggestions:First, If your goal is to simulate and explore the dynamics and control of an inverted N link pendulum, it is significantly easier if you use Wolfram System Modeler (either from within Mathematica or using it stand-alone). In SystemModeler, this double pendulum is already modeled as an example using six built-in components and connecting them. You can easily invert the pendulum and add a cart (There are some examples of a single link inverted pendulum as well). This would give you a simulation and 3D animation of the links. Additionally, you can export the equations of motion to Mathematica for manipulation. (including a linearized state space model)Second, In the blog post, the author linearizes the model around equilibrium and creates his model from StateSpaceModel[]. He also could have generated the nonlinear model with NonlinearStateSpaceModel[]. Why can't you use NonlinearStateSpaceModel or StateSpaceModel since you have the same equations?Regards,Neil
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