Hopping robot - Stance

Engine model operating under idle

Model description: 

The dynamic engine model, with parameters for a 1.6 liter, 4-cylinder fuel injected engine, is a two-input/two-output system, given by the following differential equations: $\dot{P}=k_P(\dot{m}_{ai}-\dot{m}_{ao}),$ where $k_p=42.40$

$\dot{N}=k_N(T_i-T_L),$ where $k_N=54.26$

$\dot{m}_{ai}=(1+0.907\theta+0.0998\theta^2)g(P)$

$g{P}= \begin{cases} 1, & P<50.6625 \\ 0.0197(101.325P - P^2)^{\frac{1}{2}}, &P \geq 50.6625 \end{cases}$

$\dot{m}_{ao} = -0.0005968N-0.1336P+0.0005341NP+0.000001757NP^2$

$m_{ao} = \dot{m}_{ao}(t-\tau)/(120N), \ \tau=45/N$

$T_i = -39.22+325024m_{ao}-0.0112\delta^2+0.635\delta+(0.0216+0.000675\delta)N(2\pi/60)-0.000102N^2(2\pi/60)^2$

$T_L = (N/263.17)^2+T_d$.

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TitleNeurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks
Publication TypeJournal Article
Year of Publication1994
AuthorsPuskorius, G.V., and Feldkamp L.A.
JournalIEEE Transactions on Neural Networks
Volume5
Issue2
Start Page279
Pagination279-297
Date Published1994
ISSN1045-9227
Accession Number4685633
Keywordsfiltering and prediction theory, Kalman filters, nonlinear control systems, Nonlinear dynamical systems, recurrent neural nets
AbstractAlthough the potential of the powerful mapping and representational capabilities of recurrent network architectures is generally recognized by the neural network research community, recurrent neural networks have not been widely used for the control of nonlinear dynamical systems, possibly due to the relative ineffectiveness of simple gradient descent training algorithms. Developments in the use of parameter-based extended Kalman filter algorithms for training recurrent networks may provide a mechanism by which these architectures will prove to be of practical value. This paper presents a decoupled extended Kalman filter (DEKF) algorithm for training of recurrent networks with special emphasis on application to control problems. We demonstrate in simulation the application of the DEKF algorithm to a series of example control problems ranging from the well-known cart-pole and bioreactor benchmark problems to an automotive subsystem, engine idle speed control. These simulations suggest that recurrent controller networks trained by Kalman filter methods can combine the traditional features of state-space controllers and observers in a homogeneous architecture for nonlinear dynamical systems, while simultaneously exhibiting less sensitivity than do purely feedforward controller networks to changes in plant parameters and measurement noise.
DOI10.1109/72.279191

Continuous flow stirred tank reactor

Model description: 

Coupled nonlinear differential equations describing a process involving a continuous flow stirred tank reactor are given by

$$\begin{align*} \dot{C}_1 &= -C_1u + C_1(1-C_2)e^{C_2/\Gamma} \\ \dot{C}_2 &= -C_2u + C_1(1-C_2)e^{C_2/\Gamma}\dfrac{1+\beta}{1+\beta-C_2}. \end{align*}$$

In these equations, the state variables $C_1$ and $C_2$ represent dimensionless forms of cell mass and amount of nutrients in a constant volume tank, bounded between zero and unity. The control $u$ is the flow rate of nutrients into the tank, and is the same rate at which contents are removed from the tank. The constant parameters $\Gamma$ and $\beta$ determine the rates of cell formation and nutrient consumption; these parameters are set to $\Gamma$= 0.48 and $\beta$ = 0.02 for the nominal benchmark specification.

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Model order: 

2

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Publication details: 

TitleNeurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks
Publication TypeJournal Article
Year of Publication1994
AuthorsPuskorius, G.V., and Feldkamp L.A.
JournalIEEE Transactions on Neural Networks
Volume5
Issue2
Start Page279
Pagination279-297
Date Published1994
ISSN1045-9227
Accession Number4685633
Keywordsfiltering and prediction theory, Kalman filters, nonlinear control systems, Nonlinear dynamical systems, recurrent neural nets
AbstractAlthough the potential of the powerful mapping and representational capabilities of recurrent network architectures is generally recognized by the neural network research community, recurrent neural networks have not been widely used for the control of nonlinear dynamical systems, possibly due to the relative ineffectiveness of simple gradient descent training algorithms. Developments in the use of parameter-based extended Kalman filter algorithms for training recurrent networks may provide a mechanism by which these architectures will prove to be of practical value. This paper presents a decoupled extended Kalman filter (DEKF) algorithm for training of recurrent networks with special emphasis on application to control problems. We demonstrate in simulation the application of the DEKF algorithm to a series of example control problems ranging from the well-known cart-pole and bioreactor benchmark problems to an automotive subsystem, engine idle speed control. These simulations suggest that recurrent controller networks trained by Kalman filter methods can combine the traditional features of state-space controllers and observers in a homogeneous architecture for nonlinear dynamical systems, while simultaneously exhibiting less sensitivity than do purely feedforward controller networks to changes in plant parameters and measurement noise.
DOI10.1109/72.279191

Generic nonlinear system 2

Model description: 

$$\begin{align*} x_1(k+1) &= 0.9x_1(k)\sin{[x_2(k)]} + \left(2 + 1.5 \dfrac{x_1(k)u_1(k)}{1+x_1^2(k)u_1^2(k)}\right)u_1(k) + \left(x_1(k) + \dfrac{2x_1(k)}{1+x_1^2(k)}\right)u_1(k)\\ x_2(k+1) &= x_3(k)(1+\sin{[4x_3(k)]}+ \dfrac{x_3(k)}{1+x_3^2(k)}\\ x_3(k+1) &= (3 + \sin{[2x_1(k)]})u_2(k)\\ y_1(k)&=x_1(k)\\ y_2(k)&=x_2(k) \end{align*}$$

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3

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TitleAdaptive control of nonlinear multivariable systems using neural networks
Publication TypeConference Paper
Year of Publication1993
AuthorsNarendra, K.S., and Mukhopadhyay S.
Conference NameProceedings of the 32nd IEEE Conference on Decision and Control, 1993.
Date Published12/1993
PublisherIEEE
Conference LocationSan Antonio, TX
ISBN Number0-7803-1298-8
Accession Number4772091
Keywordsadaptive control, multivariable systems, neural nets, nonlinear systems
AbstractIn this paper we examine the problem of control of multivariable systems using neural networks. The problem is discussed assuming different amounts of prior information concerning the plant and hence different levels of complexity. In the first stage it is assumed that the state equations describing the plant are known and that the state of the system is accessible. Following this the same problem is considered when the state equations are unknown. In the last stage the adaptive control of the multivariable system using only input-output data is discussed in detail. The objective of the paper is to demonstrate that results from nonlinear control theory and linear adaptive control theory can be used to design practically viable controllers for unknown nonlinear multivariable systems using neural networks. The different assumptions that have to be made, the choice of identifier and controller architectures and the generation of adaptive laws for the adjustment of the parameters of the neural networks form the core of the paper
DOI10.1109/CDC.1993.325299

Hopping robot - Flight

Model description: 

$$\dot{x}=A_Fx+b_F\tau+e_F(x)$$

with

$A_S = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & -K \dfrac{m_n}{\beta_F} & -C \dfrac{m_n}{\beta_F} \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -K \dfrac{m_{bnt}}{\beta_F} & -C \dfrac{m_{bnt}}{\beta_F} \\ \end{bmatrix},\\ x = \begin{bmatrix} z\\ \dot{z}\\ p\\ \dot{p} \end{bmatrix}, b_F = \dfrac{\eta}{\beta_{F}r} \begin{bmatrix} 0\\ m_n\\ 0\\ m_{bn} \end{bmatrix},\\ e_F(x)=\dfrac{1}{\beta_F} \begin{bmatrix} 0\\ \alpha(m_{bnt}g + f_{fF} + m_n(m_ng - k(s_0 - l_0) - f_p))\\ 0\\ -m_nf_{fF} - m_{bnt}(k(s_0-l_0) + f_a)\\ \end{bmatrix}. $

$z$ Body Height
$p$ Actuator Length
$\tau$ Motor Torque
$\theta$ Motor angle, $\theta = p/r$
$s$ Spring Length
$m_b$ 9.5kg Upper Leg Mass
$m_n$ 0.25kg Ball Nut Mass
$m_t$ 0.5kg Toe Mass
$k$ 400 N/m Spring Constant
$F_p$ 5.0N Leg Dry Friction
$F_z$ 1.5N Planarized Dry Friction
$F_a$ 0N Ball Screw Dry Friction
$c$ 5.5Ns/m Spring Viscous Friction
$\hat{\tau}$ 1.78Nm Stall Torque
$\hat{\omega}$ 2800RPM Max Speed
$\eta$ 0.95 Ball Screw Efficiency
$s_0$ 0.608m Spring Rest Length
$l_0$ 0.595m Maximum Leg Length
$J$ 2.7$\times$10$^{-4}$kgm$^2$ Motor Inertia
$\alpha$ 0.34kgm $J/r^2+m_n$
$\mu$ 0.05 $m_t/m_{bnt}$

Type: 

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4

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Attachment: 

Publication details: 

TitleDesign, modeling and control of a hopping robot
Publication TypeConference Paper
Year of Publication1993
AuthorsRad, H., Gregorio P., and Buehler M.
Conference NameProceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems '93, IROS '93.
Date Published06/1993
PublisherIEEE
Conference LocationYokohama
ISBN Number0-7803-0823-9
Accession Number5050001
Keywordslegged locomotion
AbstractThe authors report progress towards model based, dynamically stable legged locomotion with energy efficient, electrically actuated robots. The present the mechanical design of a prismatic robot leg which is optimized for electrical actuation. A dynamical model of the robot and the actuator as well as the interaction with ground is derived and validated by demonstrating close correspondence between simulations and experiments. A new continuous, and exactly implementable open loop torque control algorithm is introduced which stabilizes a limit cycle of the underlying fourth order intermittent robot/actuator/environment dynamics
DOI10.1109/IROS.1993.583877

Hopping robot - Stance

Model description: 

$$\dot{x}=A_Sx+b_S\tau+e_S(x)$$

with

$A_S = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -K\dfrac{J}{\beta_Sr^2} & -C\dfrac{J}{\beta_Sr^2} & K\dfrac{J}{\beta_Sr^2} & C\dfrac{J}{\beta_Sr^2} \\ 0 & 0 & 0 & 1 \\ K\dfrac{m_b}{\beta_s} & C\dfrac{m_b}{\beta_s} & -K\dfrac{m_b}{\beta_s} & -C\dfrac{m_b}{\beta_s} \\ \end{bmatrix},\\ x = \begin{bmatrix} z\\ \dot{z}\\ p\\ \dot{p} \end{bmatrix}, b_S = \dfrac{\eta}{\beta_{S}r} \begin{bmatrix} 0\\ m_n\\ 0\\ m_{bn} \end{bmatrix},\\ e_S(x)=\dfrac{1}{\beta_s} \begin{bmatrix} 0\\ \alpha(ks_0 - m_{bn}g- f_{fS} + m_n(m_ng - ks_0 - f_a)\\ 0\\ 0\\ \end{bmatrix}. $

$z$ Body Height
$p$ Actuator Length
$\tau$ Motor Torque
$\theta$ Motor angle, $\theta = p/r$
$s$ Spring Length
$m_b$ 9.5kg Upper Leg Mass
$m_n$ 0.25kg Ball Nut Mass
$m_t$ 0.5kg Toe Mass
$k$ 400 N/m Spring Constant
$F_p$ 5.0N Leg Dry Friction
$F_z$ 1.5N Planarized Dry Friction
$F_a$ 0N Ball Screw Dry Friction
$c$ 5.5Ns/m Spring Viscous Friction
$\hat{\tau}$ 1.78Nm Stall Torque
$\hat{\omega}$ 2800RPM Max Speed
$\eta$ 0.95 Ball Screw Efficiency
$s_0$ 0.608m Spring Rest Length
$l_0$ 0.595m Maximum Leg Length
$J$ 2.7$\times$10$^{-4}$kgm$^2$ Motor Inertia
$\alpha$ 0.34kgm $J/r^2+m_n$
$\mu$ 0.05 $m_t/m_{bnt}$

Type: 

Form: 

Model order: 

4

Time domain: 

Linearity: 

Attachment: 

Publication details: 

TitleDesign, modeling and control of a hopping robot
Publication TypeConference Paper
Year of Publication1993
AuthorsRad, H., Gregorio P., and Buehler M.
Conference NameProceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems '93, IROS '93.
Date Published06/1993
PublisherIEEE
Conference LocationYokohama
ISBN Number0-7803-0823-9
Accession Number5050001
Keywordslegged locomotion
AbstractThe authors report progress towards model based, dynamically stable legged locomotion with energy efficient, electrically actuated robots. The present the mechanical design of a prismatic robot leg which is optimized for electrical actuation. A dynamical model of the robot and the actuator as well as the interaction with ground is derived and validated by demonstrating close correspondence between simulations and experiments. A new continuous, and exactly implementable open loop torque control algorithm is introduced which stabilizes a limit cycle of the underlying fourth order intermittent robot/actuator/environment dynamics
DOI10.1109/IROS.1993.583877

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