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Dynamics and Control of a Spherical Rob ot with an Axisymmetric Pendulum Actuator
arXiv:1511.02655v1 [math.DS] 9 Nov 2015
Tatyana B. Ivanova Elena N. Pivovarova ,
Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia

Abstract This pap er investigates the p ossibility of the motion control of a ball with a p endulum mechanism with non-holonomic constraints using gaits -- the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.

Intro duction
For the last ten years an enormous amount of research has been devoted to the dynamics and control of such vehicles as single wheel robot and spherical robot moving due to changes in the position of the center of mass (see, for example, [1, 6, 7, 10, 11, 12, 13, 14]). Also, the possibility of controlling such systems by using other internal mechanisms, for example, rotors [3, 4], is actively studied. The interest in such systems is determined by the presence of some advantages in maneuverability over wheeled vehicles. For a detailed literature review on spherical robots with various moving mechanisms, their description and application areas see [9, 13, 14]. The motion of spherical robots moving due to pendulum oscillations is studied in [5, 1, 7, 8, 11, 12, 13, 14]. In particular, Nagai [13] considers the control of the motion of a pendulum spherical robot on an inclined plane and finds the maximum inclination angle of the plane at which the vehicle can move up the plane (a similar problem is considered in [7] for a single wheel robot). Schroll [14] addresses the problem of obstacle negotiation and finds the maximum height of the obstacle which

tbesp@rcd.ru archive@rcd.ru

1

such a robot can overcome. In [11] Kayacan et al. consider the control of the motion of a ball in a straight line and a circle for various types of controllers. In [1] the authors consider the problems of controlling a spherical robot with a pendulum drive in the case of slipping at the contact point and in the nonholonomic case. The fo cus is on determining an algorithm for an additional control to approach the given tra jectory from an arbitrary point in the case of a singular matrix defining the control. However, this investigation [1] leads to a physically strange conclusion that intro duction of a small parameter can lead to the controllability of this system, whereas vanishing of this parameter corresponds to the absence of controllability, which casts some doubts on the accuracy of the results obtained. This work is a continuation of analysis of the dynamics of a spherical shell rolling without slipping on a horizontal plane with Lagrange's top fixed at the center of the shell [5, 8]. Previously in [5] the equations of motion for a free system were obtained (the equations of motion for the system situated inside the rolling ball were also obtained in [2]), all necessary first integrals and an invariant measure were found, the reduction to quadratures is given. In [8] the free motion of the ball with Lagrange's top was considered, the stability of perio dic solutions was analyzed and the tra jectories of the contact point were constructed. This paper investigates the control led motion of a ball with a pendulum. Special attention is paid to the control of the ball using gaits -- the simplest motions (such as acceleration to a certain velo city and deceleration in a straight line and rotation through a given angle) and their combination for one oscillation of the pendulum and specific examples of such motions are shown. In the last part of the paper we consider the controlled motion of the ball along a straight line with fixed parameters according to a predetermined law of motion, indicate shortcomings of this approach and consider an algorithm for computing the control torques by example.

1

Equations of motion

We consider a spherical shell (Fig. 1) relative to a fixed reference frame (the axis O z is directed vertically downwards). Let Gs be the center of mass of the shell, Gt the center of mass of the top and let Rt = |Gs Gt | denote the distance between them. The vector n is directed along the axis of symmetry of the pendulum (here and in the sequel, the vectors are denoted by bold italic letters). Assuming that in the system of the tops principal axes its tensor of inertia is ^ = i = diag(i, i, i + j ), we can represent the kinetic energy of the system as [5, 8]: T= 1 MV 2 + I 2
2

Figure 1: Spherical shell with an axisymmetric pendulum fixed at its center.

+

1 mv 2 + i 2 + j ( , n )2 , 2

where V and are the linear velo city of the center of the shell and the shells angular velo city, M and I stand for its mass and moment of inertia, v and are 2

the linear velo city of the center of mass of the top and its is the mass of the top. In this paper we will investigate the controlled motion of torque can be generated by the engine which is installed at of the pendulum to the ball and sets the pendulum and, motion. Let Q be the torque generated by the engine. Then the change of the angular momentum relative to of the shell can be written as d T = I = Ro k в N o - Q, dt

angular velo city, and m the system. The control the point of attachment accordingly, the ball in Gs and the momentum

d T = M V = N o + N t + M gk , dt V

(1 )

where N o and N t are the reaction forces acting on the shell at the contact point Qo and the point of attachment of the top Gs . For the top relative to its center of mass we have d T = i + j ( , n )n = Rt n в N t + Q, dt (2 ) d T = mv = mg k - N t , dt v The linear velo city of the center of mass of the top is determined by the relation v = V + Rt в n = Ro k в + Rt в n . The linear velo city of the center of mass of the shell V is related with the angular velo city of the shell by the no-slip condition at the contact point Qo : V = rs = Ro k в , where r s is the radius vector of the contact point. The evolution of the vector n can be found from the equation n = в n. Eliminating the reaction forces No and Nt from Eqs. (1) and (2), we obtain the equations of controlled motion for the spherical shell with the axisymmetric top fixed at its geometrical center:
2 J + mRo k в ( в k ) - mRo Rt k в ( в n ) = mRo Rt k в ( в n ) - Q, 2 i + mRt n в ( в n ) - mRo Rt n в ( в k ) =

(3 )

(4 )

= -j ( , n )n - n = в n,

2 mRt

n в ( в n ) + mg Rt n в k + Q,

2 2 where J = diag(I + M Ro , I + M Ro , I ), k = (0, 0, 1)T .

3

Differentiating Eq. (3) with respect to time, we obtain the expression for : 1 1 = V вk = a в k, Ro Ro (5 )

where a is the acceleration of the contact point. Our goal is to determine a control torque Q such that the contact point (and thus the center of the ball) moves according to the specified law of motion rs (t) = = (x(t), y (t), 0)T . The velo city of the center of mass V (t) = (x(t), y (t), 0)T and its acceleration a(t) = (x(t), y (t), 0)T are prescribed functions as well. Ё Ё By prescribing the law of motion of the contact point on a plane r s (t) from (4) and (5), we obtain a system of nine equations for the pro jections of the vectors n , , and Q which can be represented as Fy = A, (6 )

where y = ( , n , Q)T , F is the matrix whose elements depend on and n ; A is the vector function depending on , n , and . For the system (6) to have a solution, i.e., for a controlled motion to be possible, it is necessary that there exist the inverse matrix F-1 , i.e., that the condition det F = 0 be satisfied, which requires the fulfilment of the condition (see also [1])
2 i + mRt > mRo Rt .

(7 )

The condition (7) can be satisfied by cho osing the corresponding geometrical characteristics of the system. For definiteness we cho ose as the pendulum a thin disk suspended on a massless ro d, whose parameters satisfy the condition (7): the radius of the disk is Rd = 0.92Ro and the length of the ro d is Rt = 0.25Ro . Note that this system is similar to a vehicle rolling without friction on the inner surface of a spherical shell. Thus, to determine the control torque Q(t), it is necessary to solve the system (6) of differential equations with the corresponding initial conditions. However, by prescribing only the law of motion rs (t), the system cannot start a new maneuver, for example, for changing the motion direction when bypassing an obstacle. In addition, it is necessary that the ball stop at the end point of a tra jectory, and this requires that the pendulum be in the lower position and the velo city and acceleration be equal to zero at the final instant of time. As a rule, it is very difficult to satisfy such conditions beforehand, since this substantially complicates the determination of the function r s (t) for the entire tra jectory of motion. Another approach to the control in maneuvering along a general tra jectory is to use gaits. This approach implies that each motion must be performed for one oscil lation of the pendulum, which is the necessary condition for a new motion to start. By combining such motions we can get any complex tra jectory (which is useful, for example, for bypassing an obstacle). In this paper we will consider both approaches -- the control using gaits (Section 2) and the motion control with fixed parameters (such as acceleration, Section 3). For ease of use, we will write all the equations of motion in dimensionless form. For this we take the mass of the pendulum m as the unit of mass, the quantity t0 =
i+j as the unit of time and the quantity x0 = g t2 as the unit of length, 0 m g Rt

4

where g is the free-fall acceleration, i.e., in the equations of motion we make the changes t t, t0 x x, x0 M M, m t0 , t0 , Q t2 0 Q. mx2 0

Also, to abbreviate some of our forthcoming formulae, we intro duce the following notation: 2 2 i0 = i + Rt , I0 = I + (1 + M )Ro .

2

The control using gaits and their connection

We represent the vector n directed along the axis of symmetry of the pendulum as n = (sin cos , sin sin , cos )T , where is the angle of deviation of the pendulum from the vertical and is the angle between the axis O x and the direction of oscillation of the pendulum (Fig. 2). To determine Q(t), we will specify the angle of deviation of the pendulum from the vertical such that at the start and the end of a maneuver the pendulum is in the lower position: (t, , T ) = sin
2

Figure 2: Determination of the angles and .

t , T

(, T )|t=0 = (, T )|t=T = 0.

(8 )

where is an as yet unknown parameter defining the amplitude of oscillation, T is the specified time of one oscillation of the pendulum which is equal to the time of one maneuver. To determine the parameter corresponding to the specified change of velo city, it is necessary to express from the system (4) using (5) and (8) the acceleration a(t, , T ) on which the additional condition is imposed
T

V (, T ) =
0

a(t, , T ) dt.

(9 )

Integrating (9) for different values of the parameter and the perio d of oscillations T , we obtain the dependence V (, T ) (quadric surface), from which we find the parameter by cho osing the required velo city and the duration of the maneuver T. Then, knowing the value , we can explicitly determine the acceleration a(t, , T ) and express the control torques from equations (4). We will consider the described algorithm in the specific cases: acceleration of the ball to a given velo city in a straight line and rotation for one oscillation of the pendulum.

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2.1

Acceleration in a straight line
the axis O x along the direction of motion. It is obvious that during the a straight line the pendulum will oscillate in one plane, in this case in O xz , therefore, the vector n directed along the axis of symmetry of the can be written as n = (sin , 0, cos )T ,

We direct motion in the plane pendulum

where is the angle of deviation of the pendulum from the vertical specified by the expression (8). Only three of nine equations of motion (4) are nontrivial:
2 I0 2 - Ro Rt (2 cos - 2 sin ) + Q2 = 0, i0 2 - Ro Rt 2 cos + Rt sin - Q2 = 0, = 2 .

(1 0 )

The value of 2 is determined from (5): a(t) 2 = - , Ro 1 = 3 = 0 (1 1 )

Substituting (11) into (10), we obtain the expression for acceleration of the ball in the form a(t, , T ) = Ro Ё (Ro Rt cos - i0 ) + Ro Rt 2 sin - Rt sin , I0 - Ro Rt cos (1 2 )

Ё where , and are explicit functions of time and the parameters and T defined f ro m (8 ). Numerically integrating (9) using (12), we obtain the surface illustrated in Fig. 3(a). This surface V (, T ) is antisymmetric relative to the change - (this can be easily shown by substituting the function (t, 1 , T ) into (12) in the explicit f o rm (8 )). In addition, Vmax as T 0, i.e., the quicker the oscillation of the pendulum o ccurs, the more the velo city increases. Figure 3(b) shows sections of the surface for different values of T . From these graphs we can see that at least two values of correspond to given V and T :

(a )

(b)

Figure 3: (a) The surface V (, T ). (b) Section formed by the intersection of the surface V (, T ) with the planes T = 5 (solid line) and T = 1 (dashed line). 6

at = 1 the pendulum performs an oscillation by a smaller angle than at = 2 (because of this the velo city varies non-monotonically at = 2 ). Cho osing the specific value of , for example 1 , substituting (12) into the equations of motion (10) and numerically integrating them, we obtain the control torques in the form Q1 = Q3 = 0, a(t, , T ) Ё Q2 = I0 + Ro Rt (cos - sin 2 ). Ro Thus, for acceleration in a straight line it is necessary to generate a control torque which is orthogonal to the direction of motion and to the plane of oscillations of the pendulum. Example 1. We consider the acceleration of the V1 = 0.5 for the time interval T = 5. The value of responds to such a motion (see Fig. 3(b)). Since the pendulum is in the lower position, the initial conditio n = (0, 0, 1)T , = (0, 0, 0)T , ball from rest to the velo city the parameter = 0.83 corball moves from rest and the ns have the form V = (0, 0, 0)T .

r = (0, 0, 0)T ,

Substituting the computed value into the equations of motion (10) and numerically integrating them (with the specified initial conditions), we obtain the corresponding control Q = (0, Q2 , 0)T for such acceleration (Fig. 4). Figure 4 also shows the time-dependence of non-zero components of the vector n , velo city V and acceleration a(t) of the ball and angular velo city of the pendulum . As seen in the figure, the ball has gained the speed V = 0.5 for the time interval T = 5 moving further with constant velo city. The pendulum has executed one full oscillation and returned to the initial condition.

Figure 4: Time dependence of the vectors n , , Q, as well as the velo city and acceleration of the ball by acceleration for one oscillation of the pendulum.

7

To stop the ball, it is necessary to cause the pendulum to perform an oscillation in the opposite direction with the same amplitude and for the same time interval (i.e. = -0.83 and T = 5). At the end of the maneuver the pendulum is in the lower vertical position as during acceleration.

2.2

Motion with change of the direction

For simplicity we assume that at the initial instant of time the ball moves along the axis O x with some constant velo city Vs . We consider a motion where the pendulum executes one oscillation in an arbitrarily predetermined direction (at an angle to the direction of motion, see Fig. 2). The ball deviates from the initial tra jectory by some angle (see Fig. 5). The vector n directed along the symmetry axis of the pendulum can be written as

Figure 5: Vs is the initial velo city of motion, V is the change of the velo city, n = (sin cos , sin sin , cos )T , Vf is the velo city of the ball after termination of a mawhere is the angle of deviation of the pendulum neuver from the vertical, is the angle between the initial directio.n of motion of the ball and the direction of oscillation of the pendulum. The equations of motion (4) in pro jections onto the axes of the fixed co ordinate system can be written as 1 I0 - Ro Rt (1 cos - 1 sin ) + Q1 = 0, 2 I0 - Ro Rt (2 cos - 2 sin ) + Q2 = 0, i0 1 - Ro Rt 1 cos - Rt sin sin - Q1 = 0, i0 2 - Ro Rt 2 cos + Rt sin cos - Q2 = 0, = 2 cos - 1 sin . The derivatives of the angular velo cities of the ball have the form V2 a2 (t) 1 = = , Ro Ro V1 a1 (t) 2 = - =- , Ro Ro 3 = 0. (1 3 )

We can represent the accelerations a1 (t) and a2 (t) as a1 (t) = a(t, , T ) cos , a2 (t) = a(t, , T ) sin , (1 4 )

where a(t, , T ) is defined by the expression (12) and is the function of time and parameters and T and is defined by the expression (8) as before. Differentiating Eq. (3) with respect to time using (13) and (14) yields Ё r = Ro k в = Ro -2 , 1 , 0
V
T

= a(t)s,

where s = (cos , sin , 0)T = V is the unit vector (constant for an one maneuver) along which the velo city changes (see Fig. 5). Since is the angle between the initial direction of motion and the direction of oscillation of the pendulum, the following proposition holds. 8

Proposition 1. The velocity of the bal l changes in the direction of oscil lation of the pendulum. If = 0, we obtain acceleration of the ball in a straight line considered in Section 2.1. Now we consider another case -- oscillation at the angle = to the initial direction of motion. Then the vector 2 directed along the symmetry axis of the pendulum can be represented as n = (0, sin , cos )T . The equations of motion (4) can be rewritten as Figure 6: Change of the velo city of the ball.

2 I0 1 - Ro Rt (1 cos + 1 sin ) + Q1 = 0,

i0 1 - Ro Rt 1 cos - Rt sin - Q1 = 0, = -1 . As in the previous case, we specify the angle o the vertical in the form (8). The derivatives of the the form a(t) V2 = , 2 = 1 = Ro Ro

(1 5 )

f deviation of the pendulum from angular velo cities of the ball have 3 = 0. (1 6 )

The expression for the acceleration of the ball a(t, , T ) has the same form as during acceleration in a straight line (12) and the controls can be written as Q1 = - a(t, , T ) Ё I0 - Ro Rt cos - sin Ro Q2 = Q3 = 0. Thus, similarly to acceleration in a straight line, for rotation through the specified angle it is necessary to generate a control torque which is orthogonal to the plane of oscillation of the pendulum and, accordingly, to the direction of the vector of change of the velo city. Differentiating Eq. (3) with respect to time and using (12) and (16), we obtain Ё r = Ro k в = Ro -2 , 1 , 0
T 2

,

(1 7 )

= (0, a(t, , T ), 0)T ,

(1 8 ) ection of through 0 and T and O y , rotation

i.e., the velo city changes only in the direction of the axis O y -- in the dir oscillation of the pendulum. Our goal is to find a value of the parameter at which a rotation a given angle is performed. To this end we integrate Eq. (18) between and compute the pro jections of the velo city V1 and V2 onto the axes O x respectively, at the final instant of time, which are related with the angle of in the absolute co ordinate system by the relation (see Fig. 6) = arctg 9 V2 . V1

Changing the parameters and T , we obtain the dependence (, T ) shown in Fig. 7(a). The function (, T ) is also antisymmetric relative to the plane = 0, and max as T 0.
2

Figure 7(b) shows sections of this surface for different values of T . As seen from the graphs, at least two values of correspond to the given and T as in the previous case.

(a )

(b)

Figure 7: (a) The surface (, T ). (b) Section formed by the intersection of the surface (, T ) with the planes T = 5 (solid line) and T = 1 (dashed line). Substituting the computed value of the parameter , for example 1 , into the equations of motion (15) and numerically integrating them using (12), we obtain the controls (17). Example 2. We consider the rotation of the ball through the angle = 40 for the time T = 5 with the initial conditions in the form n = (0, 0, 1)T , = (0, 0, 0)T , r = (0, 0, 0)T , V = (Vs , 0, 0)T ,

where Vs is the initial velo city. We cho ose the initial velo city such that the change of velo city V and the parameter are the same as in the previous example, i.e., Vs = V ctg = 0.6. The tra jectory of such a motion is shown in Fig. 8. Since all parameters are similar to those in the previous example, all functions have the form shown in Fig. 4 up to the change of variables n1 n2 , 2 -1 , Vx Vy and Q2 -Q1 .

Figure 8: The tra jectory of motion by rotation through the angle = 40, Vs = 0.6.

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3

Motion with fixed parameters

In this section we will consider a control which prescribes the motion with constant acceleration along a given straight line (or a given curve in the more general case). Such a control is used, for example, in usual vehicles, i.e., in fact it is defined in a bo dy-fixed reference frame. As will be shown below, in this case such an approach has some shortcomings which stem from the conservatism of the resulting system. We demonstrate by a specific example an algorithm for determining the control torque. For this, we consider the uniformly accelerated motion of the ball in a straight line along the axis O x under the law x(t) = where a0 = const is the given acceleratio Assume that the ball rolls without using (3) we obtain 1 (t) = 0, a0 t2 , 2

n of the ball. spinning, i.e., 3 = 0. Similarly to (11)

a0 2 (t) = - , Ro i.e., during acceleration along the axis O x the pendulum performs oscillations only in the plane O xz , therefore, the vector n has the form n = (sin , 0, cos )T , where is the angle of deviation of the pendulum from the vertical, which is the unknown function of time. The angular velo city of the pendulum has the form 2 = . Substituting the resulting expressions into the equations of motion, we find the control torques which ensure uniformly accelerated motion in a straight line along the axis O x Q1 = Q3 = 0, a0 Ё Q2 = I0 + Ro Rt cos - sin Ro

2

and the equation for determining the dependence of the angle : Ё = a0 I0 - a0 Ro Rt cos - Ro Rt Ro sin 2 + sin Ro (i0 - Ro Rt cos ) . (1 9 )

In addition to the geometrical integral n 2 = 1, this system admits another integral of motion quadratic in angular velo city C=- Ro RR (i - Ro Rt cos )2 2 + Ro Rt i0 - o t cos (cos - a0 sin ) + 20 2 - a0 Ro Rt I0 sin ,

22 Ro Rt + a0 I0 i0 + 2

which is an analog of an integral of generalized energy in a uniformly accelerated reference frame. 11

Figure 9 shows a phase portrait of the system (19) on the plane (, ) for uniformly accelerated motion of the ball in a straight line with the acceleration a0 = 0.1. For the given system parameters, there are two fixed points on the phase plane one of which corresponds to a stable equilibrium position (at = 0.38, = 0 of center type), while the other is an unstable saddle point (at = 2.56 and = 0).

Figure 9: Phase portrait of the system for uniformly accelerated motion of the ball in a straight line. As seen from Fig. 9, there exists a single tra jectory corresponding to a state of rest at the initial instant of time (it passes through the point (0, 0) -- bold line). Perio dic motion along this tra jectory is ensured by the perio dic control torque. Figure 10 shows the time dependence of the vectors n , and Q for the initial conditions (0) = (0) = 0. From these graphs we can see that the pendulum performs oscillations in the plane O xz , and the vector of the control torque changes perio dically and is directed along the axis O y .

Figure 10: Time dependence of non-zero components of the vectors n , , Q when the ball moves in a straight line with the constant acceleration a0 = 0.1 and the initial conditions = = 0. If at the initial instant of time we define the angle of deviation corresponding to the stable fixed point, then the ball will move with uniform acceleration. The deviation of the pendulum by a constant angle will be maintained by a constant value of the control torque. To any other (arbitrary) initial conditions there correspond closed perio dic tra jectories whose realization requires constantly applying 12

the perio dical control, which is inconvenient for the user. That is, it is difficult to manually maintain the motion with constant acceleration. In addition, a significant shortcoming of this control metho d is the difficulty of switching to another motion mo de at any instant of time (for example, if the velo city reaches the given value), because it may happen that the pendulum is not in the lower position. The authors are grateful to A. V. Borisov, I. S. Mamaev and A. A. Kilin for fruitful discussions and useful remarks.

References
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