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Case Study #6: Robotic Manipulator Control Design

  • Rush D. RobinettIII
  • David G. Wilson
Part of the Understanding Complex Systems book series (UCS)

Abstract

Chapter 11 presents the design of nonlinear controllers for a two-link robot based on Hamiltonian Surface Shaping and Power Flow Control (HSSPFC). HSSPFC is demonstrated to be an extension of controlled Lagrangians, energy-balancing, and energy-shaping by developing necessary and sufficient conditions for stability of a class of nonlinear systems, Hamiltonian natural systems, based on the recognition that the Hamiltonian is stored exergy. The nonlinear dynamic stability constraint is shown to be equivalent to the Melnikov number for heteroclinic orbits. HSSPFC is used to design nonlinear regulator and tracking controllers with defined stability boundaries including limit cycles. Also, the minimum energy state controller of energy-balancing is demonstrated to be a maximum entropy state controller based on HSSPFC. Numerical simulations are presented for the tracking controller design.

Keywords

Power Flow Heteroclinic Orbit Tracking Controller Rigid Body Mode Tracking Position Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

11.1 Introduction

Many real-world problems require enhanced performance while ensuring stability of nonlinear systems. In particular, the aerospace community is constantly striving toward higher performance fighter aircrafts that are intrinsically unstable and nonlinear such as the X-29 with a forward swept wing canard configuration. The X-29 is statically unstable (and dynamically unstable) without a stability augmentation systems (SAS) [106]. The X-29 SAS was designed with linear control tools and gain-scheduling. This solution limited the X-29’s performance since the present day researchers recognize the value and are attempting to utilize highly nonlinear behavior to enhance performance of future aircraft including unmanned combat air vehicles (UCAVs). Clearly, there is a need to develop nonlinear control design tools that determine and take advantage of the stability boundaries of nonlinear systems by reducing the conservativeness of current state-of-the-art tools.

The goal of developing necessary and sufficient conditions for nonlinear systems has been an area of intense research for many years. The sufficient conditions for stability are well defined in terms of the Lagrange–Dirichlet stability theorem [107, 108], Lyapunov’s stability theorems [51, 95, 109], and the absolute stability problem (passivity theorems) [95]. In fact, many of these tools [110] are making progress in developing constructive control law design procedures such as controlled Lagrangians [111], energy-shaping [112], and energy-balancing [112, 113].

Attempts to find the necessary conditions for stability, finding the on-set of instability, include the sufficient conditions for an unstable system, Lyapunov’s converse theorems [107, 108] and the inversion of the Lagrange–Dirichlet theorem [107, 108]. The limitations of these theorems are defining the transition from stability to instability and the requirements of conservative systems from the class C 2 or more restrictive [107, 108] to invert the Lagrange–Dirichlet theorem.

Chapter 11 defines the transition from stability to instability in the context of Lyapunov and dynamic stability (balanced power flow: a limit cycle) and in the context of static stability by utilizing the inversion of the Lagrange–Dirichlet theorem for systems of class C 2. The transition from stability to instability is defined in terms of the shape of the Hamiltonian surface and the power flow (work rate) by extending controlled Lagrangians [111], energy-shaping [112], and energy-balancing [112, 113] with exergy/entropy control [13] and static and dynamic stability [37]. The inversion of the Lagrange–Dirichlet theorem is used to formalize the concepts of static and dynamic stability in a nonlinear context. These extensions are a direct result of recognizing that the Hamiltonian is stored exergy and defines the accessible phase space of the system, the application of the Second Law of Thermodynamics to the power flow to determine the trajectory across the Hamiltonian surface, and the utilization of static and dynamic stability concepts to define a two-step design process.

This chapter is divided into four sections. Section 11.2 defines the equations of motion and their relationship to HSSPFC design. Section 11.3 develops a tracking controller utilizing HSSPFC and presents numerical results. Finally, in Sect. 11.4 the chapter results are summarized with concluding remarks.

11.2 Evaluation of the Equations of Motion

This section gives an illustrative example to demonstrate how to apply HSSPFC for a nonlinear mechanical system with gyroscopic or centripetal and Coriolis acceleration terms [66] that are normally associated with MIMO systems. Some of the early work performed by the authors and associated with single-axis systems are given in [30, 60, 114, 115, 116]. The equations of motion are derived with Lagrange’s equations and evaluated with the Hamiltonian and its time derivative, power flow.

11.2.1 Two-Link Robot Model

Consider a planar two-link manipulator (see reference [117] for details) with two revolute joints in the vertical plane as shown in Fig. 11.1. Let m i and L i be the mass and length of link i, r i be the distance from joint (i−1) to the center of mass of link i, as indicated in the figure, and I i be the moment of inertia of link i about the axis coming out of the page through the center of mass of link i. Referring to Fig. 11.1 the positions of each mass m i , i=1,2, are given by
$$\left\{\begin{array}{@{}c@{}}x_{c_{1}} \\y_{c_{1}}\end{array}\right\} = \left\{\begin{array}{@{}c@{}}r_1 \cos q_1 \\r_1 \sin q_1\end{array}\right\}$$
and
$$\left\{\begin{array}{@{}c@{}}x_{c_{2}} \\y_{c_{2}}\end{array}\right\} = \left\{\begin{array}{@{}c@{}}L_1 \cos q_1 + r_2 \cos(q_1 + q_2) \\L_1 \sin q_1 + r_2 \sin(q_1 + q_2)\end{array}\right\} .$$
The velocities are determined by differentiating the positions and resulting in
$$\mathbf{v}_{c_{1}} = \left\{\begin{array}{@{}c@{}}-r_1 \dot{q}_1 \sin q_1\\r_1 \dot{q}_1 \cos q_1\end{array}\right\}$$
and
$$\mathbf{v}_{c_{2}} = \left\{\begin{array}{@{}c@{}}-L_1 \dot{q}_1 \sin q_1 - r_2 (\dot{q}_1 + \dot{q}_2 ) \sin(q_1 + q_2)\\L_1 \dot{q}_1 \cos q_1 + r_2 (\dot{q}_1 + \dot{q}_2 ) \cos(q_1 + q_2)\end{array}\right\}.$$
The planar two-link robot presented in Fig. 11.1 has the following kinetic energy, potential energy, Lagrangian, and Hamiltonian:
$$\begin{array}{@{}l}\displaystyle\mathcal{T} = \frac{1}{2}m_1 \mathbf{v}_{c_{1}} \cdot \mathbf{v}_{c_{1}}+ \frac{1}{2} I_1 \dot{q}_1^2 +\frac{1}{2}m_2 \mathbf{v}_{c_{2}} \cdot \mathbf{v}_{c_{2}}+ \frac{1}{2} I_2 (\dot{q}_1 + \dot{q}_2 )^2, \\[2mm]\mathcal{V} = m_1 g y_{c_{1}} + m_2 g y_{c_{2}}, \\[1.5mm]\mathcal{L} = \mathcal{T} - \mathcal{V}, \\[1.5mm]\mathcal{H} = \mathcal{T} + \mathcal{V},\end{array}$$
(11.1)
where g is the gravitational constant. The equations of motion are generated from Lagrange’s equation
$$\frac{d}{dt} \biggl( \frac{\partial{\mathcal{L}}}{\partial\dot{q}_i}\biggr)- \frac{\partial\mathcal{L}}{\partial q_i} = Q_i,\quad i=1,2,$$
(11.2)
and the virtual work is
$$\delta W = \sum_{i=1}^2 Q_i \delta q_i \quad\mbox{for}\ Q_i =u_i + f_i (q_i,\dot{q}_i),$$
(11.3)
where u i is the ith control torque, and f i is the ith frictional torque. Performing the indicated Lagrangian operations results in the dynamic equations of motion for the planar two-link robot
$$\mathbf{M}(\mathbf{q}) \ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q},\dot{\mathbf{q}}) \dot{\mathbf{q}}+ \mathbf{F} (\mathbf{q},\dot{\mathbf{q}}) + \mathbf{G}(\mathbf{q})= \mathbf{u}, $$
(11.4)
where \(\mathbf{q} = [q_{1} \; q_{2} ]^{T} \in\mathcal{R}^{2}\) is the angular position vector, \(\mathbf{u} = [ \tau_{1} \; \tau_{2}]^{T} \in\mathcal{R}^{2}\) is the control torque, and
$$\mathbf{M}(\mathbf{q}) = \left[\begin{array}{@{}c@{\quad}c@{}}\bar{m}_1 + \bar{m}_2 + 2 \bar{m}_3 \cos q_2 & \bar{m}_2 + \bar{m}_3\cos q_2 \\[2mm]\bar{m}_2 + \bar{m}_3 \cos q_2 & \bar{m}_2\end{array}\right],$$
$$\mathbf{C}(\mathbf{q}, \dot{\mathbf{q}} ) = \left[\begin{array}{@{}c@{\quad}c@{}}-\bar{m}_3 \dot{q}_2 \sin q_2 & -\bar{m}_3 (\dot{q}_1 + \dot{q}_2 )\sin q_2 \\[2mm]\bar{m}_3 \dot{q}_1 \sin q_2 & 0\end{array}\right],$$
$$\mathbf{F}(\mathbf{q},\dot{\mathbf{q}}) = \left[\begin{array}{@{}c@{}}c_1 \dot{q}_1 + c_{NL_{1}} \operatorname{sign}(\dot{q}_1) \\[2mm]c_2 \dot{q}_2 + c_{NL_{2}} \operatorname{sign}(\dot{q}_2)\end{array}\right],$$
$$\mathbf{G}(\mathbf{q}) = \left[\begin{array}{@{}c@{}}\bar{m}_4 g \cos q_1 + \bar{m}_5 g \cos(q_1 +q_2) \\[2mm]\bar{m}_5 g \cos(q_1 + q_2)\end{array}\right],$$
$$\begin{array}{@{}l}\bar{m}_1 = m_1 r_1^2 + I_1 + (m_2 + m_p) L_1^2, \\[2mm]\bar{m}_2 = m_2 r_2^2 + I_2 + m_p L_2^2, \\[2mm]\bar{m}_3 = m_2 L_1 r_2 + m_p L_2^2, \\[2mm]\bar{m}_4 = m_1 r_2 + (m_2 + m_p)L_1, \\[2mm]\bar{m}_5 = m_2 r_2 + m_p L_2,\end{array}$$
where M(q) is a positive definite matrix, \(\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\) is a skew symmetric matrix, \(\mathbf{F}(\mathbf{q},\dot{\mathbf{q}})\) is a linear and nonlinear friction vector with friction coefficients c i and \(c_{NL_{i}}\), i=1,2, G(q) is a gravitational force vector given by \(\mathbf{G}(\mathbf{q}) = \partial\mathcal{V} / \partial\mathbf{q}\), and m p is the payload mass.
Fig. 11.1

MIMO planar robot model. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

11.2.2 Evaluation of the Hamiltonian Surface Shaping

The first step in the HSSPFC design process is to recognize that the system is constrained to move on the Hamiltonian surface, the accessible phase space, which can be projected onto the phase plane.

Following the processes in Chap.  4, a statically neutral system is not typically identified separately, but it has important physical significance for mechanical systems. For the two-link robot model operating in a horizontal plane (G(q)=0), the system is statically neutral stable with two rigid body modes, \(\mathcal{V}(\mathbf{q}) = 0\ \forall \mathbf{q}\), which leads to a singular stiffness operator and no preferred configuration. The two-link robot model is unstable without proportional feedback control. For the two-link robot operating in the vertical plane, the system is only statically stable about the robot hanging vertically straight down without feedback control. Static stability is a necessary condition for stability, but not sufficient, and it limits the performance of the system. Returning to the X-29 example, this airplane was designed to be longitudinally statically unstable, and the degree of static stability was generated with the SAS depending upon the desired level of maneuverability (dog fighting, level flight, landing).

Controlled Lagrangians [111], energy-shaping [112], and energy-balancing [112, 113] can be used to construct a feedback controller that meets the sufficient conditions for stability. However, these tools do not recognize the importance of the Hamiltonian surface. Basically, any proportional feedback controller that derives from a C 2 function (and some C 1 functions) meets the requirements of static stability and can be used to increase performance by reducing the stability margin and even driving the system unstable for a portion of the path. A simple example is where
$$\mathbf{u} = \left\{\begin{array}{@{}c@{}}- K_{P_{1}}( q_1 - q_{1_{R}} ) - K_{NL_{1}}( q_1 -q_{1_{R}})^3 \\[2mm]- K_{P_{2}} ( q_2 - q_{2_{R}} ) - K_{NL_{2}}( q_2 -q_{2_{R}})^3\end{array}\right\} .$$
(11.6)
An example of reducing the stability margin and bifurcating the statically stable equilibrium point occurs when \(K_{P_{i}}\) is changed from \(K_{P_{i}} > 0\) to \(K_{P_{i}} = 0\) and to \(K_{P_{i}} < 0\). Figure 11.2 presents the results for \(K_{P_{1}}\).
Fig. 11.2

Static stability to bifurcation of an equilibrium point. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

11.2.3 Evaluation of Power Flow

The second step in the HSSPFC design process is to identify the Hamiltonian as stored exergy, take the time derivative, and apply the Second Law of Thermodynamics in order to partition the power flow into three types: (i) the energy storage rate of change, (ii) power generation, and (iii) power dissipation [13, 37, 46, 50], and then follow the processes in Chap.  4.

A dynamically stable system is equivalent to energy-shaping [112] and energy-balancing [112, 113] except for the generator terms that do not meet passivity requirements, the line integral that is used to calculate average values of the power flows (i.e., AC power, discontinuous functions, etc.), and the balance of power generation to power dissipation subject to the power storage that leads to a limit cycle as a stability boundary [13, 37, 46].

A dynamically unstable system is equivalent to the converse of Lyapunov stability with the addition of the line integral. A dynamically neutral stable system is not typically identified separately, but it has important physical significance for mechanical systems, especially in aeroelasticity [118]. Dynamically neutral stability is the on-set of a limit cycle oscillation. Also, this equation plays an important role in determining the preservation of heteroclinic orbits [99]. The Melnikov number [99, 119] is defined as
$$L = \Delta\mathcal{H} = \int_{-\infty}^{\infty} \dot{\mathcal{H}}\,dt =\int_0^{\tau} \dot{\mathcal{H}}\,d t = 0,$$
(11.7)
which implies a zero change of energy over a heteroclinic orbit that is preserved.
For the structured robot dynamics, defined earlier, the Hamiltonian rate is
$$\dot{\mathcal{H}} = \dot{\mathbf{q}}^T [\mathbf{M} \ddot{\mathbf{q}} +\mathbf{G}(\mathbf{q})] =\dot{\mathbf{q}}^T [-\mathbf{F}(\mathbf{q},\dot{\mathbf{q}})-\mathbf{C}(\mathbf{q},\dot{\mathbf{q}}) + \mathbf{u}] .$$
It can be noted that the gyroscopic or centripetal/Coriolis terms do no work over a cycle and therefore drop out [15]. In addition, for the two-link robot model operating in a horizontal plane, G(q)=0, the time derivative of the Hamiltonian is the power flow/work rate [15] for natural systems,
$$\dot{\mathcal{H}} = \dot{\mathcal{T}} = \sum_{i=1}^2 \{ Q_i \dot {q}_i =[u_i - c_i \dot{q}_i - c_{NL_{1}} \operatorname{sign} (\dot{q}_i)] \dot{q}_i\} .$$
(11.8)
This system is statically neutral stable and dynamically unstable with no control inputs due to the rigid body modes. For a nonlinear PID regulator,
$$u_i = -K_{P_{i}}q_i - K_{P_{NL_{i}}} q_i^3 - K_{I_{i}} \int_0^t q_i\,dt- K_{D_{i}} \dot{q}_i- K_{D_{NL_{i}}} \operatorname{sign} (\dot{q}_i) .$$
The power flows can be sorted as [13, 37, 46] with storage terms
$$\int_0^{\tau_c}\dot{\mathcal{T}}\,dt + \int_0^{\tau_c} \sum_{i=1}^2\bigl[K_{P_{i}} q_i+ K_{P_{NL_{i}}} q_i^3 \bigr] \dot{q}_i \,dt = 0,$$
generator terms
$$\frac{1}{\tau_c} \int_0^{\tau_c} \sum_{i=1}^2 \biggl[-K_{I_{i}}\int_0^tq_i \,d \tau\biggr]\dot{q}_i\,dt > 0,$$
dissipator terms
$$\frac{1}{\tau_c} \int_0^{\tau_c}\sum_{i=1}^2 \bigl[-(K_{D_{i}}+c_{i}) \dot{q}_i^2- (K_{D_{NL_{i}}}+c_{NL_{i}}) \operatorname{sign}(\dot{q}_i) \dot{q}_i\bigr]\,dt < 0,$$
statically stable
$$\mathcal{V}_c(\mathbf{q}) > 0 \quad\mbox{for} \quad \mathbf{q} \neq \mathbf{0},\qquad\mathcal{V}_c(\mathbf{0}) = 0,$$
dynamically stable for subject to dynamically unstable when and dynamically neutral stable when for i=1,2.

For the two-link robot operating in the vertical plane, a feedforward computed torque controller can be used to account for the gravity terms while everything else remains the same. Note that the statically and dynamically stable controller reaches a stable equilibrium point that is a minimum energy state [112] and a maximum entropy state [13] since the dominant dissipator term is equal to the irreversible entropy production term, \(T_{0} \dot{\mathcal{S}}_{i}\).

11.3 Tracking Controller: Perfect Parameter Matching

In this section a tracking controller is designed for perfect parameter matching and to demonstrate decoupling of the DOFs for the three cases of stable, neutral, and unstable conditions [66]. The PID controller is defined for each DOF as
$$\boldsymbol{\tau} = \hat{\mathbf{M}}\ddot{\mathbf{q}}_{\mathrm{ref}}+\hat{\mathbf{C}}\dot{\mathbf{q}}_{\mathrm{ref}}-\mathbf{K}_P\tilde{\mathbf{q}} - \mathbf{K}_I\int_0^t \tilde{\mathbf{q}}\,d \tau-\mathbf{K}_D \dot{\tilde{\mathbf{q}}},$$
(11.10)
where K P , K I , and K D are the proportional, integral, and derivative diagonal matrix controller gains, respectively. The errors are given by \(\tilde{\mathbf{q}} = \mathbf{q}_{\mathrm{ref}} - \mathbf{q}\) and \(\dot{\tilde{\mathbf{q}}} = \dot{\mathbf{q}}_{\mathrm{ref}} -\dot{\mathbf{q}}\) with subscript ref equaling reference input terms.
The tracking servo control design begins with picking a Lyapunov function/Hamiltonian based on the error energy
$$V = \Delta\mathcal{H} = \frac{1}{2} \dot{\tilde{\mathbf{q}}}^T \mathbf{M} \dot {\tilde{\mathbf{q}}}+ \frac{1}{2} \tilde{\mathbf{q}}^T \mathbf{K}_P \tilde{\mathbf{q}},$$
(11.11)
which is positive definite and statically stable. The time derivative is and upon substitution of the dynamic error equations, simplifications, and assumption of perfect parameter matching (\(\hat{\mathbf{M}}= \mathbf{M}\), \(\hat{\mathbf{C}} =\mathbf{C}\)), one has
$$\dot{V} = - \dot{\tilde{\mathbf{q}}}^T \mathbf{K}_D \dot{\tilde{\mathbf{q}}}- \dot{\tilde{\mathbf{q}}}^T \mathbf{K}_I \int_0^t \tilde{\mathbf{q}}\, d \tau $$
(11.13)
that is subject to the following general necessary and sufficient conditions:
$$\begin{array}{@{}l@{\quad}l}\bigl[ T_0 \Delta\dot{\mathcal{S}}_i\bigr]_{\mathrm{ave}} \geq 0 &\mbox{positive semi-definite, always true}, \\[1.5mm]\bigl[ \Delta\dot{W}\bigr]_{\mathrm{ave}} \geq 0 & \mbox{positive\ semi-definite; exergy pumped in}.\end{array}$$
Similar arguments can be made for the corollaries as compared to the earlier subsection. The formulation is identical to the regulator problem with the exception that for the nonlinear case, additional cross-terms are accommodated by the identity \(\dot{\tilde{\mathbf{q}}}^{T} [\dot{\mathbf{M}}-2\mathbf{C}]\dot{\tilde{\mathbf{q}}} =0\).
Next, the PID tracking control in terms of exergy generation and exergy dissipation is investigated. The derivative of the Lyapunov function/Hamiltonian (11.13) yields
$$\begin{array}{@{}l}\bigl[ T_0 \Delta\dot{\mathcal{S}}_i \bigr]_{\mathrm{ave}} =\dot{\tilde{\mathbf{q}}}^T \mathbf{K}_D \dot{\tilde{\mathbf{q}}},\\[2mm]\displaystyle \bigl[ \Delta\dot{W} \bigr]_{\mathrm{ave}} =- \dot{\tilde{\mathbf{q}}}^T \mathbf{K}_I\int_0^t \tilde{\mathbf{q}}\,d \tau, \\[2.7mm]\displaystyle \bigl[ T_0 \Delta\dot{\mathcal{S}}_{\mathrm{rev}}\bigr]_{\mathrm{ave}} =\bigl[ \dot{\tilde{\mathbf{q}}}^T \mathbf{M}\ddot{\tilde{\mathbf{q}}}\bigr]_{\mathrm{ave}}+\bigl[ \dot{\tilde{\mathbf{q}}}^T \mathbf{K}_P\tilde{\mathbf{q}}\bigr]_{\mathrm{ave}} = 0 .\end{array} $$
(11.14)
The first expression in (11.14), the derivative tracking control term, is identified as dissipative terms. The second expression in (11.14), the integral tracking control term, is identified as a generative term. In the final expression in (11.14), the proportional tracking control term is identified as a reversible term along with the inertial terms.
To determine the nonlinear stability boundary from the HSSPFC design, let
$$\bigl[ \Delta\dot{W}\bigr]_{\mathrm{ave}} =\bigl[T_0 \Delta\dot{\mathcal{S}}_i\bigr]_{\mathrm{ave}} .$$
Substituting the actual terms yields the following:
$$\begin{array}{@{}l}\displaystyle\bigl[ K_{D_{1}} \dot{\tilde{q}}_1^2 \bigr]_{\mathrm{ave}} =\biggl[- K_{I_{1}} \int_o^t \tilde{q}_1 \,d \tau \dot{\tilde{q}}_1\biggr]_{\mathrm{ave}}, \\[4mm]\displaystyle\bigl[ K_{D_{2}} \dot{\tilde{q}}_2^2 \bigr]_{\mathrm{ave}}= \biggl[- K_{I_{2}} \int_o^t \tilde{q}_2 \,d \tau \dot{\tilde{q}}_2 \biggr]_{\mathrm{ave}},\end{array} $$
(11.15)
which are the nonlinear stability boundaries per DOF. To best understand how the boundary is determined, concepts and analogies from electric AC power have been introduced earlier. Essentially, when the average powerin is equivalent to the average powerdissipated over a cycle, then the system is operating at the stability boundary. Later, in the exergy and exergy rate responses for the nonlinear system, one may observe that the area under the curves for the exergy rate generation and the exergy rate dissipation are equivalent, and for the corresponding exergy responses, the slopes will be equal and opposite.
Numerical simulations are performed for three separate PID tracking control cases with the numerical values listed in Table 11.1.
Table 11.1

MIMO planar robot PID controller numerical values. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

Case No.

\(K_{P_{1}}\) (N m)

\(K_{P_{2}}\) (N m)

\(K_{D_{1}}\) (N m/s)

\(K_{D_{2}}\) (N m/s)

\(K_{I_{1}}\) (N m/s)

\(K_{I_{2}}\) (N m/s)

1

250

100

7

2

8.8

2.75

2

250

100

7

2

880

275

3

250

100

3.5

0.267

880

275

The robot physical parameters are given as: link lengths; L 1=L 2=0.5 m, center of gravity locations r 1=r 2=0.25 m, moments of inertia I 1=I 2=0.1 kg m2, and masses m 1=4.5 kg and m 2=2.5 kg, respectively. The detailed dynamic model is given in [117].

The reference input signal is defined as \(\mathbf{q}_{\mathrm{ref}} =\mathbf{A}_{\mathrm{in}}\sin\boldsymbol{\omega}_{\mathrm{in}} t\) with \(\mathbf{A}_{\mathrm{in}}=[1 \; 2]\) rad and \(\boldsymbol{\omega}_{\mathrm{in}} = [1 \; 2]\) rad/s for each DOF, respectively. The MIMO robot system is initially at zero. For Case 1, the tracking position along with the reversible proportional tracking control exergy and exergy rate responses are given in Fig. 11.3 for both DOFs. Note that for over each cycle, the proportional tracking control exergy rate response is zero (valid for all three cases). Variations for dissipative and generative exergy and exergy rate responses are given in Fig. 11.4 for DOF one (left) and two (right), respectively. For Case 1, passive stable tracking is observed from the decaying response and exhibits a predominately larger perturbation dissipative term with respect to the perturbation generative term. The initial transient tracking position errors converge to the reference sinusoidal inputs. For Case 2, similar responses for the tracking position errors and reversible proportional tracking control exergy and exergy rate responses are presented in Fig. 11.5 for both DOFs. The variations for the dissipative and generative exergy and exergy rate responses are illustrated in Fig. 11.6 for DOF one (left) and two (right), respectively. This case demonstrates the neutrally stable tracking boundary or where the dissipative terms are equivalent and cancel the generative terms. For Case 3, similar responses are given in Figs. 11.7 and 11.8. Case 3 presents responses that are growing exponentially, since the generative term is greater than the dissipative term.
Fig. 11.3

Case 1: Robot MIMO tracking and reversible K P exergy and exergy rate numerical results. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

Fig. 11.4

Case 1: Robot MIMO tracking exergy and exergy rate numerical results. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

Fig. 11.5

Case 2: Robot MIMO tracking and reversible K P exergy and exergy rate numerical results. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

Fig. 11.6

Case 2: Robot MIMO tracking exergy and exergy rate numerical results. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

Fig. 11.7

Case 3: Robot MIMO tracking and reversible K P exergy and exergy rate numerical results. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

Fig. 11.8

Case 3: Robot MIMO tracking exergy and exergy rate numerical results. Robinett III, R.D. and Wilson, D.G. [66], reprinted by permission of the publisher (©2006 ASME)

11.4 Chapter Summary

Chapter 11 has applied HSSPFC to a nonlinear MIMO system with gyroscopic or centripetal/Coriolis acceleration terms. HSSPFC was demonstrated to be an extension of controlled Lagrangians, energy balancing, and energy shaping by developing necessary and sufficient conditions for stability of a class of nonlinear systems, Hamiltonian natural systems, based on the recognition that the Hamiltonian is stored exergy. It was demonstrated how the nonlinear dynamic stability constraint is equivalent to the Melnikov number for heteroclinic orbits. HSSPFC was used to design nonlinear regulator and tracking controllers with defined stability boundaries including limit cycles for a two-link robot. Also, the minimum energy state controller of energy-balancing was shown to be a maximum entropy state controller based on HSSPFC.

References

  1. 13.
    Robinett III, R.D., Wilson, D.G.: Exergy and irreversible entropy production thermodynamic concepts for nonlinear control design. Int. J. Exergy 6(3), 357–387 (2009) Google Scholar
  2. 15.
    Junkins, J.L., Kim, Y.: Introduction to Dynamics and Control of Flexible Structures. AIAA, Washington (1993) MATHGoogle Scholar
  3. 30.
    Robinett III, R.D., Wilson, D.G.: Exergy and entropy thermodynamic concepts for control system design: slewing single axis. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, CO, August 2006 Google Scholar
  4. 37.
    Robinett III, R.D., Wilson, D.G.: Hamiltonian surface shaping with information theory and exergy/entropy control for collective plume tracing. Int. J. Systems, Control and Communications 2(1/2/3) (2010) Google Scholar
  5. 46.
    Robinett III, R.D., Wilson, D.G.: What is a limit cycle? Int. J. Control 81(12), 1886–1900 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 50.
    Robinett III, R.D., Wilson, D.G.: Collective plume tracing: a minimal information approach to collective control. Int. J. Robust Nonlinear Control (2009). doi: 10.1002/rnc.1420 MATHGoogle Scholar
  7. 51.
    Lyapunov, A.M.: Stability of Motion. Academic Press, New York (1966) MATHGoogle Scholar
  8. 60.
    Robinett III, R.D., Wilson, D.G.: Exergy and irreversible entropy production thermodynamic concepts for control design: nonlinear systems. In: 14th Mediterranean Conference on Control and Automation, Ancona, Italy, June 28–30, 2006 Google Scholar
  9. 66.
    Robinett III, R.D., Wilson, D.G.: Exergy and entropy thermodynamic concepts for nonlinear control design. In: ASME 2006 International Mechanical Engineering Congress & Exposition, Chicago, IL, November 5–10, 2006 Google Scholar
  10. 95.
    Slotine, J.-J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, Englewood Cliffs (1991) MATHGoogle Scholar
  11. 99.
    Alberto, L.F.C., Bretas, N.G.: Application of Melnikov’s method for computing heteroclinic orbits in a classical SMIB power system model. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47(7), 1085–1089 (2000) MathSciNetMATHCrossRefGoogle Scholar
  12. 106.
    Winchester, J.: Grumman X-29, X-Planes and Prototypes. Amber Books, London (2005) Google Scholar
  13. 107.
    Taliaferro, S.D.: An inversion of the Lagrange–Dirichlet stability theorem. Arch. Ration. Mech. Anal. 73, 183–190 (1980) MathSciNetMATHCrossRefGoogle Scholar
  14. 108.
    Hagedorn, P., Mawhin, J.: A simple variational approach to a converse of the Lagrange-Dirichlet theorem. Arch. Ration. Mech. Anal. 120, 327–335 (1992), Springer-Verlag MathSciNetMATHCrossRefGoogle Scholar
  15. 109.
    Willems, J.C.: Dissipative dynamical systems part I: General theory; part II: Linear systems with quadratic supply rates. Arch. Ration. Mech. Anal. 45, 321–393 (1972) MathSciNetMATHCrossRefGoogle Scholar
  16. 110.
    Kokotovic, P., Arcak, M.: Constructive nonlinear control: a historical perspective. Preprint submitted to Elsevier, August 2000 Google Scholar
  17. 111.
    Bloch, A.N., Chang, D.E., Leonard, N.E., Marsden, J.E.: Controlled Lagrangians and the stabilization of mechanical systems: potential shaping. IEEE Trans. Autom. Control 46(10), 1556–1571 (2001) MathSciNetMATHCrossRefGoogle Scholar
  18. 112.
    Ortega, R., Garcia, E.: Energy-shaping stabilization of dynamical systems. Laboratoire des Signaux et Systemes, SUPELEC, Gif-sur-Yvette, France (2004) Google Scholar
  19. 113.
    Takegaki, M., Arimoto, S.: A new feedback method for dynamic control of manipulators. Journal of Dynamic Systems, Measurement, and Control 102(119) (1981) Google Scholar
  20. 114.
    Robinett III, R.D., Wilson, D.G.: Exergy and irreversible entropy production thermodynamic concepts for control system design: regulators. In: 2006 IEEE International Conference on Control Applications (CCA), Munich, Germany, October 2006, pp. 2249–2256 (2006) CrossRefGoogle Scholar
  21. 115.
    Robinett III, R.D., Wilson, D.G.: Exergy and irreversible entropy production thermodynamic concepts for control system design: robotic servo applications. In: 2006 IEEE International Conference on Robotics and Automation, Orlando, FL, May 15–19, 2006, pp. 3685–3692 (2006) Google Scholar
  22. 116.
    Robinett III, R.D., Wilson, D.G.: Exergy and irreversible entropy production thermodynamic concepts for control system design: nonlinear regulator systems. In: The 8th IASTED International Conference on Control and Applications, Montreal, Quebec, Canada, May 24–26, 2006 Google Scholar
  23. 117.
    Qu, Z., Dawson, D.M.: Robust Tracking Control of Robot Manipulators. IEEE Press, New York (1996) MATHGoogle Scholar
  24. 118.
    Fung, Y.C.: An Introduction to the Theory of Aeroelasticity. Dover, New York (1969) Google Scholar
  25. 119.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983) MATHGoogle Scholar

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© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Sandia National LaboratoriesAlbuquerqueUSA

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