Dynamics and control of a multi-body planar pendulum

Abstract

The explicit equations of motion for a general $n$-body planar pendulum are derived in a simple and concise manner. A new and novel approach for obtaining these equations using mathematical induction on the number bodies in the pendulum system is used. Assuming that the parameters of the system are precisely known, a simple method for its control that is inspired by analytical dynamics is developed. The control methodology provides closed-form nonlinear control and makes no approximations/linearizations of the nonlinear system. No a priori structure is imposed on the controller. Globally, asymptotic Lyapunov stability is achieved along with the minimization of a user-provided control cost at each instant of time. This control methodology is then extended to include uncertainties in the parameters of the system through the use of an additional continuous controller. Simulations showing the simplicity and efficacy of the approach are provided for a 10-body pendulum system whose model is only known imprecisely. The ease with which the uncertain system can be controlled to move from any initial state to various final so-called inverted configurations is demonstrated.

Appendix

In this section, the proofs for the results regarding the additional controller used in Sect. 4 are provided. For convenience, we recall the following.

The equation of motion of the controlled nominal system is

$$\begin{aligned} M(\theta )\ddot{\theta }=Q(\theta ,\dot{\theta })+Q^{C}(t). \end{aligned}$$

(49)

The equation of motion of the controlled actual system is

$$\begin{aligned} M_a (\theta _a )\ddot{\theta }_a =Q_a (\theta _a ,\dot{\theta }_a )+Q^{C}(t)+Q^{u}(\theta _a ,\dot{\theta }_a ). \end{aligned}$$

(50)

In the above equation, the additional (generalized) control force $Q^{u}$ is computed using the expression

$$\begin{aligned} Q^{u}=-\beta (s/\varepsilon ) \end{aligned}$$

(51)

where $\varepsilon$ is a small positive number, $s$ is the sliding variable

$$\begin{aligned} s:=\dot{e}_a +ke_a , \quad k>0, \end{aligned}$$

(52)

and $\beta$ is a positive number satisfying the condition

$$\begin{aligned}&\beta \ge \frac{\left\| {\delta \ddot{q}} \right\| +k\left\| {\dot{e}_a } \right\| }{\lambda _{\min } }, \forall t,\hbox { where } \nonumber \\&\quad \lambda _{\min }:=\min \{eigenvalues\hbox { of }M_a^{-1} \}. \end{aligned}$$

(53)

In Eq. (53), $\delta \ddot{q}$ is a quantity defined as,

$$\begin{aligned} \delta \ddot{q}:=M_a^{-1} (Q_a +Q^{C})-M^{-1}(Q+Q^{C}), \end{aligned}$$

(54)

and $\left\| {\hbox {.}} \right\|$ represents $L_2$ norm of a vector.

Result 1: The additional control force $Q^{u}$ given by

$$\begin{aligned} Q^{u}=-\beta (s/\varepsilon ) \end{aligned}$$

(55)

where $\varepsilon$ is a small positive number and $\beta$ is a positive number satisfying the condition given in Eq. (53) ensures that the controlled actual system

$$\begin{aligned} M_a (\theta _a )\ddot{\theta }_a =Q_a (\theta _a ,\dot{\theta }_a )+Q^{C}(t)+Q^{u}(\theta _a ,\dot{\theta }_a ). \end{aligned}$$

(56)

stays with in the region $\Omega _\varepsilon$ defined by

$$\begin{aligned} \Omega _\varepsilon :=\left\{ {s\in R^{n}|\left\| s \right\| \le \varepsilon } \right\} . \end{aligned}$$

(57)

Proof

Noting the definition of the sliding manifold in Eq. (52), its derivative with respect to time is,

$$\begin{aligned} \dot{s}(t)=\ddot{e}_a +k\dot{e}_a . \end{aligned}$$

(58)

Upon differentiating the tracking error $e_a (t)=\theta _a (t)-\theta (t)$ twice, we have

$$\begin{aligned} \ddot{e}_a (t)=\ddot{\theta }_a (t)-\ddot{\theta }(t). \end{aligned}$$

(59)

Using the equations of motion of the controlled nominal system (Eq. 49) and the controlled actual system (Eq. 56), Eq. (59) becomes

$$\begin{aligned} \ddot{e}_a= & {} M_a^{-1} (Q_a +Q^{C})-M^{-1}(Q+Q^{C})+M_a^{-1} Q^{u}\nonumber \\= & {} \delta \ddot{q}+M_a^{-1} Q^{u}. \end{aligned}$$

(60)

The last equality above is obtained from the definition of $\delta \ddot{q}$ in Eq. (54). Thus, the time derivative of the sliding manifold can be simplified using Eq. (60) as,

$$\begin{aligned} \dot{s}(t)=\ddot{e}_a +k\dot{e}_a =\delta \ddot{q}+M_a^{-1} Q^{u}+k\dot{e}_a . \end{aligned}$$

(61)

Considering the Lyapunov function

$$\begin{aligned} V_a =\frac{1}{2}s^{T}s, \end{aligned}$$

(62)

its rate of change along the trajectories of the dynamical system is given by

$$\begin{aligned} \dot{V}_a= & {} s^{T}\dot{s}\nonumber \\= & {} s^{T}(\delta \ddot{q}+M_a^{-1} Q^{u}+k\dot{e}_a )\nonumber \\= & {} s^{T}(\delta \ddot{q}-M_a^{-1} \beta \left( {\frac{s}{\varepsilon }} \right) +k\dot{e}_a ). \end{aligned}$$

(63)

Observing that $s^{T}M_a^{-1} s\ge \lambda _{\min } \left\| s \right\| ^{2}$, we have

$$\begin{aligned} \dot{V}_a\le & {} \left\| s \right\| \left\| {\delta \ddot{q}} \right\| +k\left\| s \right\| \left\| {\dot{e}_a } \right\| -\beta \lambda _{\min } \frac{\left\| s \right\| ^{2}}{\varepsilon } \nonumber \\= & {} \left\| s \right\| \left( {\left\| {\delta \ddot{q}} \right\| +k\left\| {\dot{e}_a } \right\| -\beta \lambda _{\min } \frac{\left\| s \right\| }{\varepsilon }} \right) . \end{aligned}$$

(64)

The region $\Omega _\varepsilon$ is defined such that $\left\| s \right\| \le \varepsilon$, and we have $\left\| s \right\| /\varepsilon >1$ outside $\Omega _\varepsilon$. Hence outside $\Omega _\varepsilon$, the right-hand side of Eq. (64) is strictly negative when $\beta \ge \frac{\left\| {\delta \ddot{q}} \right\| +k\left\| {\dot{e}_a } \right\| }{\lambda _{\min } }$. Since the controlled actual system starts inside the region $\Omega _\varepsilon$, it stays within this attracting region and cannot escape from it. $\square$

As pointed out in Remark 5, if the nominal system and the uncertain system do not start with the same initial conditions, then any trajectories of the controlled uncertain system that start from outside $\Omega _\varepsilon$ are globally attracted to the region $\Omega _\varepsilon$.

Result 2: If the controlled actual system is restricted to stay within the region $\Omega _\varepsilon$, the errors in tracking the nominal system are bounded by,

$$\begin{aligned} \left| {e_{a,i} } \right| \le \frac{1}{k}\varepsilon , \left| {\dot{e}_{a,i} } \right| \le 2\varepsilon , i=1,2,\cdots ,n. \end{aligned}$$

(65)

Proof

Inside the region $\Omega _\varepsilon$, $\left\| s \right\| \le \varepsilon$ and hence,

$$\begin{aligned} \left| {s_i } \right| \le \varepsilon ,i=1\cdots n. \end{aligned}$$

(66)

From the relation $s_i =\dot{e}_{a,i} +ke_{a,i}$, we get,

$$\begin{aligned} \left| {\dot{e}_{a,i} +ke_{a,i} } \right| \le \varepsilon ,i=1\cdots n. \end{aligned}$$

(67)

This inequality can be alternatively expressed as,

$$\begin{aligned} -\varepsilon \le \dot{e}_{a,i} +ke_{a,i} \le \varepsilon , \end{aligned}$$

(68)

which can further be simplified to

$$\begin{aligned} -\varepsilon -ke_{a,i} \le \dot{e}_{a,i} \le \varepsilon -ke_{a,i} . \end{aligned}$$

(69)

Considering $e_{a,i}$ as a dynamical system, if we can prove that $e_{a,i} \dot{e}_{a,i} <0$ (which is the derivative of the Lyapunov function $\frac{1}{2}e_{a,i} e_{a,i} )$ outside a region $L_\varepsilon ^i$, we can conclude that the region $L_\varepsilon ^i$ is an attracting region. Defining $L_\varepsilon ^i$ as,

$$\begin{aligned} L_\varepsilon ^i :=\left\{ {e_{a,i} \in R|\left| {e_{a,i} } \right| \le \frac{1}{k}\varepsilon } \right\} , \end{aligned}$$

(70)

there are two possible cases in which $e_{a,i}$ could lie outside $L_\varepsilon ^i$. Let us look at both of them.

Case 1: If $e_{a,i} >\frac{1}{k}\varepsilon >0$, then $\varepsilon -ke_{a,i} <0$. From Eq. (69), we then have

$$\begin{aligned} e_{a,i} \dot{e}_{a,i} \le e_{a,i} \left( {\varepsilon -ke_{a,i} } \right) <0. \end{aligned}$$

(71)

Case 2: If $e_{a,i} <-\frac{1}{k}\varepsilon <0$, then $\varepsilon +ke_{a,i} <0$. Also, $e_{a,i} <0$ and so from the left inequality in (69), we have,

$$\begin{aligned} e_{a,i} \dot{e}_{a,i} \le -e_{a,i} \left( {\varepsilon +ke_{a,i} } \right) <0. \end{aligned}$$

(72)

We note that initially $e_{a,i} =0$, and therefore, it will remain inside the region $L_\varepsilon ^i$ thereafter, or in other words, $\left| {e_{a,i} } \right| \le \frac{1}{k}\varepsilon$. From relation (67), we observe that

$$\begin{aligned} \left| {\left| {\dot{e}_{a,i} } \right| -\left| {ke_{a,i} } \right| } \right| \le \left| {\dot{e}_{a,i} +ke_{a,i} } \right| \le \varepsilon , \end{aligned}$$

(73)

which further yields

$$\begin{aligned} \left| {\dot{e}_{a,i} } \right| \le \varepsilon +\left| {ke_{a,i} } \right| \le 2\varepsilon . \end{aligned}$$

(74)

$\square$