Simultaneous stabilization and trajectory tracking of underactuated mechanical systems with included actuators dynamics

Abstract

The paper deals with a novel control algorithm for simultaneous stabilization and trajectory tracking of underactuated nonlinear mechanical systems (UNMS) with included actuators dynamics. Simultaneous stabilization and trajectory tracking refer to arbitrary chosen actuated and unactuated degrees of freedom (DOF) of the system. The proposed control approach can be applied both to the second-order nonholonomic systems and the systems with input coupling, while a general model of actuators dynamics includes electrical, pneumatic, and hydraulic drives. Control law is based on linear combination of two control signals, where the first signal is designed to separately control only actuated DOF, and second to separately control only unactuated DOF. Simulation example of rotational inverted pendulum driven by electrical DC motor is presented, showing the effectiveness of the proposed approach.

Appendices

Appendix A

A proof of the matrix inversion for matrix $\tilde{\mathbf {M}}_{a1}(\mathbf{q})$, is derived in the similar way as given in [25], and is given by the following expression:

(32)

where $\mathbf{I}_{n_{a}\times n_{a}} \in\mathbb{R}^{n_{a}\times n_{a}}$ is identity matrix, and $\mathbf{0}_{n_{u}\times n_{a}} \in\mathbb {R}^{n_{u}\times n_{a}}$ is zero matrix. Since the matrix M(q) is positive definite matrix, and matrix

$$\begin{bmatrix} \mathbf{I}_{n_a\times n_a} \\ -\mathbf {M}_{u2}^{-1}(\mathbf{q})\mathbf{M}_{u1}(\mathbf{q})\end{bmatrix} $$

has full column rank n _a , it follows that matrix $\tilde{\mathbf {M}}_{a1}(\mathbf{q})$ has full rank n _a , i.e. it is invertible.

A proof of matrix inversion for matrix $\tilde{\mathbf{M}}_{u2}(\mathbf {q})$, is derived in the similar way as given in [25], and is given by the following expression:

(33)

Since the inertia matrix M(q) is positive definite matrix, and matrix

$$\begin{bmatrix} -\mathbf{M}_{a1}^{-1}(\mathbf{q})\mathbf{M}_{a2}(\mathbf {q}) \\ \mathbf{I}_{n_u\times n_u}\end{bmatrix} $$

has full column rank n _u , it follows that matrix $\tilde{\mathbf {M}}_{u2}(\mathbf{q})$ has full rank n _u , i.e., it is invertible.

Appendix B

Relation between the function and the input vector u using expressions from (9) and (12) is given by

(34)

where is a vector function, at k=n _a , while $\gamma_{ac}^{k}(\mathbf{x}_{ac},\mathbf{x}_{q})$ is a scalar function. The matrix of partial derivatives is defined by the following expression:

(35)

where, for instance, the matrix element in (35) is in position (1,2) defined as shown:

$$ \frac{\partial\gamma_{ac}^1(\mathbf{x}_{ac},\mathbf{x}_q)}{\partial \mathbf{x}_{ac}^2}= \frac{\partial\gamma_{ac}^1(\mathbf{x}_{ac},\mathbf{x}_q)}{\partial x_{ac1}^2}+ \frac{\partial\gamma_{ac}^1(\mathbf{x}_{ac},\mathbf{x}_q)}{\partial x_{ac2}^2}+\cdots+ \frac{\partial\gamma_{ac}^1(\mathbf{x}_{ac},\mathbf{x}_q)}{\partial x_{acj}^2}$$

(36)

Here, $j=n_{ac}^{2}$, i.e. j represents a number of the state space variables of the actuator 2, while the number 2 from $j=n_{ac}^{2}$ denotes a mark for actuator and does not mean squaring of n _ac .

Appendix C

A pneumatic actuator with proportional spool valve consists of a pneumatic cylinder with piston and a proportional spool valve, as shown in Fig. 5. A mathematical model of the pneumatic actuator [6, 21] consists of model of the piston dynamics, model of the fluid pressure dynamics and model of the proportional valve spool dynamics, while for kth actuator is a vector of the state space variables defined as follows:

$$ \mathbf{x}_{ac}^k= \left [\begin{array}{c}x_{ac1}^k \\x_{ac2}^k \\x_{ac3}^k \\x_{ac4}^k\end{array} \right ]= \left [ \begin{array}{c}p_1 \\p_2 \\x_s \\\dot{x}_s\end{array}\right ]$$

(37)

where state variables represent: p ₁—air pressure in the left cylinder chamber, p ₂—air pressure in the right cylinder chamber, x _s —spool displacement and $\dot{x}_{s}$—spool velocity. In modeling, this actuator has two additional state variables, x _c —cylinder piston displacement, and $\dot{x}_{c}$—cylinder piston velocity, but these variables are replaced by the state variables ($\mathbf{x}_{qa1}^{k},\mathbf{x}_{qa2}^{k}$) of the mechanical system. This replacement is a consequence of an assumption of the rigid connection between the actuator and mechanical system, as explained in Sect. 2.3. For that case, variables x _c i $\dot{x}_{c}$ do not belong to the vector $\mathbf{x}_{ac}^{k}$.

Fig. 5

Schematic representation of the pneumatic cylinder-valve system

3.1 C.1 Cylinder piston dynamics

A cylinder piston dynamics is described by ordinary second-order differential equation, as shown by the following expression:

$$ F^k=\underbrace{A_1p_1-A_2p_2}_{ \gamma_{ac}^k(\mathbf {x}_{ac}^k,\mathbf{x}_{qa1}^k,\mathbf{x}_{qa2}^k)}-\underbrace {M_c}_{M_{ac}^k} \underbrace{\ddot{q}_a^k}_{\dot{x}_{qa2}^k}-\underbrace{F_{\mathrm{dis}_{ac1}}\bigl(q_a^k,\dot{q}_a^k\bigr)}_{F_{\mathrm{dis}_{ac1}}^k(\mathbf {x}_{qa1}^k,\mathbf{x}_{qa2}^k)}$$

(38)

where M _c , A ₁, A ₂ are positive constants, representing mass of the piston, active surface area of the piston for chambers 1 and 2, respectively. The expression $F_{\mathrm{dis}_{ac1}}(x_{c},\dot{x}_{c})$ represents friction force (which is in Fig. 5 denoted as F _dis) dependent on the state variables of the cylinder. The expression A ₁ p ₁−A ₂ p ₂ represents a force acting on the piston and caused by air pressures p ₁ and p ₂, while F ^k is a force produced by a kth pneumatic actuator. A rigid connection between state variables of the actuator and mechanical system is described by expressions $x_{c}=q_{a}^{k}$ and $\dot{x}_{c}=\dot{q}_{a}^{k}$. For this case, a transmission ratio is N=1. An underlined representation in the equation (38) is a state space form, where state variables are x _ac1=p ₁ and x _ac2=p ₂.

3.2 C.2 Fluid pressure dynamics

Air pressures in cylinders’ chambers are defined as follows:

$$ \begin{aligned} &\overbrace{\dot{p}_1}^{\dot{x}_{ac1}}=\overbrace{\frac{\kappa RT}{V_D+V_0+A_1q_a^k}\dot{m}_1(x_s,p_1)-\frac{\kappa A_1}{V_D+V_0+A_1q_a^k}p_1\dot{q}_a^k}^{f_{ac1}^k(\mathbf {x}_{ac}^k,\mathbf{x}_{qa1}^k,\mathbf{x}_{qa2}^k)}\\&\underbrace{\dot{p}_2}_{\dot{x}_{ac2}}=\underbrace{\frac{\kappa RT}{V_D+V_0-A_1q_a^k}\dot{m}_2(x_s,p_2)+\frac{\kappa A_2}{V_D+V_0-A_2q_a^k}p_2\dot{q}_a^k}_{f_{ac2}^k(\mathbf {x}_{ac}^k,\mathbf{x}_{qa1}^k,\mathbf{x}_{qa2}^k)}\end{aligned} $$

(39)

where index 1 refers to the left chamber and index 2 refers to the right chamber of the cylinder.

Parameters κ, R, T, V _D , V ₀, A ₁, A ₂ are constants greater the zero, representing the specific heat ratio, ideal gas constant, temperature, inactive cylinder volume at the and of stroke and admission port, half of the active cylinder volume, and the piston effective area, respectively. Variables p _i and $\dot{m}_{i}$, for i∈{1,2} represent air pressures and mass flow rates in cylinder chambers. Mass flows $\dot{m}_{i}$, for i∈{1,2} are expressed as functions of pressures and spool displacement x _s (see Fig. 6), and described by following equations:

(40)

(41)

where: C _f , C ₁ and C ₂ are constants greater the zero, A _v (x _s ) is the valve effective area for input/exhaust paths which depends on spool displacement, p _s is the pressure in the air reservoir, p _e is the exhaust pressure, p _cr is critical pressure value.

Fig. 6

Schematic representation of valve spool dynamics

3.3 C.3 Valve spool dynamics

$$ M_s\ddot{x}_s+c_s\dot{x}_s+F_{\mathrm{dis}_{ac3}}(x_s,\dot{x}_s)+2k_sx_s=F_c$$

(42)

where M _s , k _s , c _s are constants greater the zero, representing the spool coil assembly mass, the spool springs constant, and the viscous friction coefficient, respectively. The expression $F_{\mathrm{dis}_{ac3}}(x_{s},\dot{x}_{s})$ represents the friction force (which is in Fig. 6 denoted as F _dis) dependent on the valve’s state variables. The input F _c represents the force produced by the coil, and it can be linearly described as a function of electric current F _c =K _fc i _c or voltage. The equation (42) can be expressed in a state space form as follows:

(43)

(44)

where state variables are x _ac3=x _s and $x_{ac4}=\dot{x}_{s}$.

The whole model of the pneumatic actuator (39), (43), (44), and (38), with general form presented in Sect. 2.3, is given by the following expressions:

(45)

Note: A relative degree of this pneumatic actuator, according to the output F ^k, is r=3. A first time derivation of $\gamma_{ac}^{k}$ contains x _s , a second time derivative of $\gamma_{ac}^{k}$ contains $\dot{x}_{s}$, while a third time derivative of $\gamma_{ac}^{k}$ contains $\ddot{x}_{s}$ and it contains the input u ^k according to (44).