In this note, we will propose a control strategy for non-
I n t r o d u c t i o nWe will propose a new control strategy for a class of nonholonomic systems by defining the timestate control form. This approach allows us to control non-holonomic systems which can not be transformed to chained form[2,3]. We will apply this control strategy to the position control of multi-trailer systems. In [5], we will apply this strategy to a space robot.holonomic systems using the time-state control form.
State E q u a t i o n of Non-Holonomic SystemsMany non-holonomic systems, such as multi-trailer systems and space robots, are modeled in the following m input nonlinear system.where x E R" is a state, U = ( u I , . . . , u~)~ E Rm is a input with m < n, and g,(x) is a n-dimensional vector valued function with {g1(0),g2(O),..-,gm(O)} linearly independent. As Brockett showed in [l], the system (1) can not be stabilized with any continuous static state feedbacks, i.e. there do not exist any continuous functions r,(z) such that U, = r,(z) will stabilize the system. Thus, we can not use conventlonal controller design methods. Example: Consider the trailer in Fig.1, which is one of the well known non-holonomic systems[2,4]. If we assume that there are no slips in any wheels, then we have where (12, y2) represent the position of the trailer; 81 and 82 represent the orientation of the tractor and the trailer, respectively; L1 and L2 are wheel bases of the tractor and the trailer, res ectively; and z is the tractor's distance along the reafpath. This system has two inputs: Z (the velocity of the tractor) and Q (the steering andefined w = i and w = i t a n a , and the angular velocity of the tractor, respectively. Since g1(0,0,0,0) = (1, O,O, O)T and g2(0,0,0,0) = (0, O , l / L I , O)T, this state equation satisfies the condition of the system (1). Thus, this system can not be stabilized with any continuous static state feedbacks.
Control S t r a t e g yIn this section, we will propose a control strate y for a class of non-holonomic systems (1). Firstly, we wi% define the time-state control form, and transform the system (1) 0-7803-1 968-0/94$4.0001994 IEEE into this form. Then we will propose a control strategy which makes the state sufficiently close to the origin.
Time-State Control FormTime-state control form is the following.--0p1-t. * * + fm-l(t)pm-l, (3) -= dt h(€,r)pm. dr dr (4) This system consists of two state equations. The state equation (3) is called the state control part, whose state is E E R"-' and time scale is r rather than the actual time scale i. This part is controlled by m -1 inputs y1,. :. , pm-1. We assume that its first order approximation IS controllable, i.e. the system = A € + b l p l + . ' . + b m -l p m -l ,is a controllable linear system.The other state equation (4) is called the time control part. Its state is r E RI, i.e. the time scale of the state control part (3). It is controlled by single input pm. In other words, the time control part controls the time scale r of the state control part. There is a ...
