A Simplified Forward Gait Control for a Quadruped Walking Robot

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  • A Simplified Forward Gait Control for a Quadruped Walking Robot

    Daniel J. Pack and A. C. Kak Robot Vision Laboratory

    1285 EE Building Purdue University

    West Lafayette, IN 47907-1285

    ABSTRACT-In this paper, we present a simplified forward gait for a quadruped walking robot. The proposed gait is a straight line, periodic, monotonically forward (SLPMF) gait and can easily be used for adaptive gait control, requiring only simple modifications. We show that given a support pattern, meaning a polygon generated by connecting the feet positions that are in contact with the ground, only certain sequences of leg movements will generate the SLPMF gait. We also intro- duce a useful method to determine how to preserve stability of a quadruped robot during the motions called for by an SLPMF gait. Another noteworthy aspect of our paper is the discussion on the leg design from a hardware standpoint; the design per- mits independent joint control.

    1. Introduction

    Study of walking robots started in the mid 60's when an ini- tial prototype of a quadruped robot was built and tested by General Electric Corporation[lG]. Since then much work in this area by various researchers has resulted in the uncovering of many problems. At one end of the spectrum, these p rob lems deal with low-level but critically important issues such as gait control, force-feedback control for terrain-adapted foot- placement, stability, etc. At the other end, we have problems of dynamic control, the incorporation of environment sensing, collision avoidance, goal attainment and others. To cite some of the literature that has appeared to date, Song and Chen[22], Hirose and Umetani[8], Bares and Wittakertl], and Choi and Song[3] have addressed the problems dealing with low-level mo- tion control for leg coordination. Hiroee[7] and Waldron et al.[24] have dealt with the problem of overall mechanical design of walking robots. For higher level issues the reader is referred to Sekiguchi, Nagata, and Asakawa[l9], Ooka et al.[17], and Brooks[2].

    In our laboratory, we are currently engaged in building a quadruped robot. The main goal of this research effort is to an- alyze the problems that arise when low-level control strategies are integrated with high-level reasoning and planning modules. As we showed in the context of wheeled mobile robots [ I l , 131, the integration of low and high level control is of fundamental importance in order to imbue a robot with "intelligent" behav- ior. While our research on quadruped robots is driven by the desire to study problems involved with such integration, this

    paper will deal with just one specific issue that we addresscd and resolved during the course of our research: the issue of gait control.

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    I d m A -

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    U U Figure 1: Diagram of our quadruped walking robot.

    The specific research we have reported in this paper is moti- vated by our credo that the complexity of a high-level controller can be reduced if the low-level control strategies are as simple as possible but, of course, adequate for the tasks intended for the robot. (The reader may see some parallels between this and the RISC approach to computing.) To simplify the low- level aspects of gait control, we have made certain reasonable assumptions about the motions that a quadruped robot is al- lowed to execute. Basically, a quadruped robot can either move its body in some direction while all its feet are stationary and in contact with the ground, or the robot may lift one of its legs and move the leg in some direction with or without the body in simultaneous motion. In this paper, we assume that at any instant of time the robot is allowed only one type of motion: the robot may either just move its body while the feet stay stationary; or, keeping its body stationary, the robot may just move one of the legs for foot-placement that is more advan- tageous for subsequent motions. These assumptions help us

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  • devise a robust and computationally inexpensive gait control algorithm that ensures the robot will stay stable during all its motions.

    The paper is organized as follows. In section 2, we discuss the mechanical design of the robot. In section 3, we then propose a forward gait that will be referred to as a straight line, periodic, monotonically forward (SLPMF) walking gait. We analyze the problem of determining the sequence of free legs for such a gait. Section 4 presents our conclusions.

    2. Mechanical Design of the Robot

    Figure 1 shows the overall configuration of the quadruped robot under construction in our lab. The main body of the robot is an oval-shaped platform populated with different types of hardware and sensors. Attached to the body are four legs in the manner shown. The mechanical design of the legs will be discussed later. In what follows immediately, we will start with a justification for why we chose four for the number of legs.

    Figure 2: Each leg consists of three links. The two joints on the left ore revolute joints while the third joint is a linear joint.

    At this time, we are interested in control strategies that are dependent solely on kinematic considerations. Evidently, dynamic considerations are of paramount importance and underlie the design of control strategies for the hopping robot by Raibert, Brown, and Chepponis[l8], biped-walking robots by Furusho and Masubuchi[4], Kajita, Yamaura, and I

  • Figure 3: A photograph of one of the legs on the robot.

    lead screw attached to the shaft of the motor turns in the base marked in Fig. 2 for the purpose of lifting or dropping the foot. Also note that hydraulic actuation as used in the Planetary Rover could not possibly be used in the manner we have for positioning the foot.

    The electrical motors used for all the links on a leg are of the stepper variety. In comparison with DC servo motors, modern stepper motors allow far easier software development as control can be accomplished in an open-loop mode without damaging the system. A typical DC servo motor will servo with respect to a specified set point and if the attainment of that set point is impeded, either the current or the voltage applied to motor will increase continuously to levels that might compromise safety. On the other hand, if the commanded motion of a stepper motor is impeded for whatever reason, the motor controller will continue to send pulses to the motor but a t a voltage no different from before. Of course, using stepper motion in open- loop control compromises precision due to lack of feedback. This we have rectified by the inclusion of encoders shown in Fig. 2. Shown in Fig. 3 is a photograph of one leg.

    While the robot is still in the development phase, we have chosen to mount all the twelve motor drivers, three for each ieg, off board. The connection between the chassis containing the motor drivers and the host computer consists of an RS232 serial line. Eventually, all of t,liis hardware, together with a microprocessor based control board will be mounted on top of the robot, much in the same manner we have done for our wheeled mobile robot [11, 131.

    3. SLPMF Walking Gait

    First a few definitions are in order. A leg will be called a sup- porting leg if its foot is in contact with the ground and exerts pressure on the ground. A leg that is lifted clear off the ground will be called a free leg. A foot placement will be considered reachable if it does not violate any of the mechanical and kine- matic constraints on the kinematic chain of the associated leg. Shown in Fig. 4 are the reachability regions for all four legs.

    As is perhaps intuitively obvious, three generic motions of a quadruped walking robot are: 1) The robot moves its body while the feet stay in contact with the ground - there is obvi- ously a limit on the size of such motions. 2) the robot lifts one of the legs and moves the foot forward for a new placement while the body stays stationary. And 3) the robot lifts one of the legs and both the body and the free leg move forward. As discussed by McGhee and Frank [12], a cyclic gait may be syn- thesized by combining these generic motions. (We must hasten to add that for primitive forward motions of the creeping kind, as exhibited by toy animal robots for kids, it is not necessary to synthesize a gait using the generic motions we just listed; the primitive kinds of walks can be created by moving the forward two legs only incrementally so that the c.0.g. of the robot is not disturbed overly and just sliding the rear legs forward as necessary.)

    4

    Figure 4: Reachable regions for each leg of the quadruped robot: Region i i s the reachable region for leg a.

    We will now present the forward periodic walking gait ad- vanced by McGhee and Frank [12]. We will refer to this gait as the McGhee and Rank gait in subsequent discussions. In this gait, forward movement of the robot is attained by simul- taneously moving the body and one of the legs that is free. For this to occur, all joints of the robot, except for joint 3 on each of the three supporting legs (this joint is connected to the lidi in touch with the ground), must be controlled.

    In Fig. 5, the corners of the polygons represent the feet and the polygon itself the support pattern of the robot. The figure shows the successive support patterns involved in the McGhee and Frank gait. Note that the concept of a support pattern, first put forward by Hildebrand [6] , is a convenient way to

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  • show each step of a gait cycle. In the figure, specified leg numbers represent the supporting legs while a missing number between one and four indicates the free leg. The arrow shows the direction of the motion; its length is proportional to the distance traveled by the center of the.body.

    4 4 l4-d 3 3

    Figure 5: Walking gait sequence proposed by McGhee and Fmnk: (a) shows initial support pattern and forward robot body motion; (b) shows leg 3 off the ground and moving to a new location. Note that unlike an our case, as the free foot moues forward, the robot body also ezecutes a forward motion simul- taneously; (e) shows leg 3 positioned on the ground and the body moving forward to allow leg 2 to be the nest free leg; in (d) leg 2 moues to a new location; (e) shows the intermediate step where leg 2 is now placed on the ground a t a new location while, simultaneously, the body moves forward also; in (f) leg 4 is lifted off the ground and placed a t a new location; in (9) leg 4 is again on the ground and the body is moved forward subsequently; finally, in (h) leg 1 is lifted and placed a t a new location which completes a gait cycle. An important point to note for this McGhee arid Fmnk gait: the body moues forward during all steps of the gait.

    If we set up our robot for the McGhee and Frank gait, we would need 9 actuator commands for each step that starts with a triangular support pattern in Fig 5; and 8 actuator commands at each step starting with a a four-sided polygon; therefore, a total of 68 commands would be needed over a gait cycle. This number of commands follows from the fact that when purely body motions are executed, that is when a four-sided support pattern moves into another four-sided support pattern, we needed coordinated motions at joints 1 and 2 for each of the four legs, requiring a total 8 such commands. Taking into account that we have four such steps in the gait cycle shown in Fig. 5, we arrive at 32 actuation commands. And, when the robot motions in a gait cycle correspond to motions from a 3-sided to a 4-sided support pattern, we require 9 actuator

    commands; of these, six will be for joints 1 and 2 of the three supporting legs and three for coordinated movement of the three links of the free leg. Since we have four such 3-sided to 4-sided transitions in the gait cycle of Fig. 5, 36 actuation commands will be required. Therefore, we conclude that a total of 68 commands would be necessary to execute all the motions in a gait cycle.

    We will now introduce a new gait control strategy that requires fewer actuation commands for the completion of one gait cycle. However, we must first introduce the notion of a S t a b i l i t y Admitting Line (SAL) that we have found useful for analyzing the static stability of the robot during a gait.

    Definition S t a b i l i t y Admitting Lines (SALS) Lines in the ground plane that either connect the supporting feet 1 and 3 or the supporting feet 2 and 4 are called stability admitting lines.

    The importance of such lines was first observed by EIirose [7]. The location of c.0.g. of a robot with respect to the SALs can determine the next appropriate action the robot needs to take. Fig. 6 shows a typical leg configuration with the SALs labeled. It is clear that the SALs always divide the plane containing the robot body into at most four different regions, labeled Quad 1, Quad 2, Quad 3, and Quad 4 in (Fig. 6(a)) and Fig. G(b) for two different support patterns.

    We will now make the reasonable assumption that at all times the positions of the four feet are known with respect to the robot body coordinate frame centered at the c.0.g. of robot. This means that the equations for the SALS can be easily con- structed for those feet that are in contact with the ground. Once the SALS are found, the robot can readily determine which quadrant contains the c.0.g.

    Figure 6: The quadrants generated by Stability Admitling Lines. The quadrants are labeled Quad 1, Quad 2, Quad 9, and Quad 4. In (a) i s shown a situation where all four quad- mnts exist while (b) shows a case where only Quad 9 and Quod 4 esist.

    If the c.0.g. is in Quad 1, the robot will not be able to use leg 1 or leg 2 as a free leg for the next motion, since otherwise the robot will lose balance and lurch forward. This problem can be remedied by first moving the robot body backwards while all the feet stay in contact with the ground. Such a motion of

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  • the body will shift the c.0.g back into Quad 3. Subsequently, the robot could plan its forward motion again and use either leg 1 or leg 2 as a free leg. If, on the other hand, the c.0.g. is in Quad 4, meaning on one side of the robot center, then it will not be possible to use leg 1 as a free leg for the next motion. Later, we will present strategies for dealing with each such situation.

    We will now present our SLPMF gait that is more efficient compared to Frank and Mcghee gait in the sense that our gait requires fewer actuation commands per gait cycle. While the manner in which we select the free leg for forward motions is same as in [12], there are two important points of departure in how we synthesize a gait. First, the location of the c.0.g. in relation to SALS plays a critical role in our work. And, second, unlike McGhee and Frank [12], we only allow the body to move forward when all four feet are in contact with the ground. When the free leg executes its motions, the body remains stationary. Although for identical joint speeds, this will mean that a robot with our gait will move slower in relation to a robot with the McGhee and Frank gait, we do not consider that a serious issue. As recent experience with systems with a large number of degrees of freedom has demonstrated, future progress will be determined more by whether or not we can discover simplified control strategies rather than by the raw overall speed. In any case, it is always possible to use more powerful motors if speed is really an issue and, at the same time, benefit from our simplified gait control.

    Figure 7: Initial configuration of the robot. Shown in (a) is the situation in which the robot c.0.g lies in Quad 1; in [b) the robot c.0.g. lies in Quad 2.

    Given all possible sequences of free legs and body movements, we first want to show that not all leg sequences can be used to generate an SLPMF walking gait. There are 24 different pos- sible sequences in which free legs can be ordered; this follows from the permutation Pi with n = r = 4. These sequences are of the form (1,2,3,4), (1,2,4,3), (1,3,4,2), etc., where the num- bers identify the legs. Among these 24 sequences, only a subset of sequences, combined with the robot body movements, are permissible for an SLPMF walking gait. The number of per- missible sequences is a function of the location of the robot c.0.g. As was mentioned earlier, if the c.0.g. is in Quad 1 of Fig. 6, it will not be possible for the robot to lift leg 1, imply- ing that sequences beginning with the number 1 would not be permissible. We now present the proposed gait assuming that the robot is initially in one of the two support patterns shown in Fig. 7. By symmetry, our discussion can be extended to

    other initial support patterns.

    Lets first consider the case of the starting support pattern on the left in Fig. 7(a). Again, as we mentioned before, for this starting configuration, the robot will not be able to lift either of the two front legs initially. Thus, only 12 possible sequences remain to be examined. Among these, only two sequences, (3,2,4,1) and (4,1,3,2), can be used since the others will either result in the robot moving backward before it goes forward or will result in a non-periodic gait. We will now show the gait corresponding to one of the two permissible sequences, (3,2,4,1).

    t

    Figure 8: For the starting support pattern shown in Fig. S(a), shown here ure the diflerent steps of the SLPMF walking gait. Shown in (a) is the initial robot configuration where leg 3 i s about to move to a new location; (b) shows leg 9 ut a new po- sition and leg 2 selected f o r the next motion; [c) shows leg 2 at the new location and the body ready to move in anticipation of the next free leg, leg 4 , as shown in (d); (e) shows the robot configuration after leg 4 has executed its motion and the se- lection of leg 1 as the next free leg; ( f ) shows leg 1 at a new location and the body ready to moue forward; (g) shows the robot body at its new location with the robot again in its initial configuration, signifying the completion of the gait cycle.

    Fig. 8 shows the steps involved in the SLPMF gait generated by the (3,2,4,1) sequence of free legs when the starting config- uration is as shown in Fig. 7(a). A circle around a particular

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  • foot indicates the leg that will be free for the next step. The precomputed sequence of free legs and body movements is 3 -+ 2 -+ move body forward -+ 4 -+ 1 -+ move body forward. In the first step, leg number 3 is selected as the free leg and moved forward such that the robot c.0.g. resides in Quad 3 or Quad 4 generated by newly computed SAL 1. Note that only one SAL needs to be updated at a time. In the next step, the leg number 2 is moved. At this point, the body needs to move such that the robot c.0.g. lies in Quad 1 or Quad 2. This allows the next free leg to be leg number 4, Fig. 8(d). During the next step, the free leg, leg number 4, needs to move forward such that the robot c.0.g. lies in Quad 2 or Quad 3 generated by the new SALS, Fig. 8(e). Once the next free leg, leg number I, is moved, the robot body needs to. be moved forward, Fig. 8(f), to restore the original body and leg configuration which ends a gait cycle.

    R

    Figure 9: For the starting support pattern shown in Fig. S(b), shown here are the diflerent steps of the SLPMF walking gait. Shown in (a) is the initial robot configuration where leg 1 is about to move to a new location; (b) shows leg 1 a t a new position and body ready to move in anticipation of the next free leg, leg 3, as shown in (c); (d) shows the robot configuration after leg 3 has executed its motion and the selection of leg 2 as the next free leg; (e) shows leg 2 a t a new location and the body ready to move forward; (f) shows the robot body a t i ts new location; and, finally, (g) shows the robot again in its initial configuration, signifying the completion of the gait cycle.

    Now consider the case where the start,ing support pattern is

    as in Fig. 7(b). Of the 24 sequences, those beginning with leg number 2 will not do, again because of stability considerations. Note, however, sequences that begin with leg 3 are now per- missible even though it seems that lifting this leg would cause the robot to lose balance. The reason is that, for the case of a sequence beginning with leg 3, the robot can move its body first, shifting the c.0.g. forward into Quad 1. Discounhg the impermissible sequences, we are left with three, (1,3,2,4), (3,4,1,2), and (4,1,3,2). In Fig. 9, we show one cycle o the gait for the sequence (1,3,2,4). The gait can be described by the following leg and body movements: 1 -+ move body forward -+ 3 -+ 2 -+ move body forward -+ 4.

    We now show one example of an impermissible sequence. Con- sider the free leg sequence (1,3,4,2). We can illustrate this case using Fig. 9. In this figure action upto frame (d) are the same provided that at the step shown in frame (d), leg 4 is the next free leg selected instead of leg number 2. At this point, the robot body needs to move forward to position its c.0.g. ahead of SAL 1 into Quad 1; this allows leg 4 to be the next free leg. Observe, however, once leg 4 is moved, wherever it may be, the next free leg can not be leg 2 since that would mean the body of the robot would have to move backward.

    Of course, if we drop the requirement that the gait be periodic or monotonically forward, we can generate a forward gait which is an end result of side walking and moving the robot c.0.g. back and forth with any sequence of body and leg movements. [This type of gait is important and in some cases necessary, but can not be an optimal straight line forward gait in which we are interested at present.]

    We will now discuss the issue of foot placement. There are two ways of determining the precise locations for placing the fcet during execution of a gait. One may first establish the gait, as we have already done, and then by using a different set of con- siderations (such as the maximization of the forward moveniritt during one gait cycle and the minimizatioii of the deviations of the body c.0.g. from a straight line motion) determine where the feet should be placed. This approach is theoretically at- tractive for obvious reasons but cannot be pursued at this time due to the nonlinearities involved in the minimization and the dimensionality of the underlying space. An opposite approach - the approach that we have adopted because it is simpler - is to assume ab initio that the the lelt and the right feet of the robot will always be placed on two parallel tracks, as showli in Fig. lO(a), ensuring that the robot body will travel along a straight line, and then determining the extent of each free leg motion for the maximization of forward progress. This raises the question of what distance to choose for the separation be- tween the two tracks. Intuitively, and we believe on theoretical grounds also, it would appear that the ideal placement of the two tracks would be as shown in Fig. 10(b) where the left track passes through the points where the left legs are anchored to the body, and the same for the right track with respect to the right legs. However, such tracks are not practical because of the hardware that is mounted on the joints between the different sections of each legs (there should be no collisions be- tween this hardware and the main body of the robot). For this reason, the tracks that we have chosen, shown in Fig. lO(a),

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  • are offset from those in Fig. 10(b) by an extent that permits the joints to move freely without running into the robot body. With the specification of the tracks, the only remaining factor that decides placement of the feet is the sequence chosen for the selection of the free legs during a gait. Currently, the se- quence that yields the largest distance forward during one gait cycle is chosen by an algorithm that is run offline.

    Figure 10: Tracks for foot placement in an SLPMF gait. Shown in (a) i s the track we have adoptedfor our robot while (b) shows ideal tracks.

    As the reader will recall, the robot executes only two differ- ent kinds of motions: either the body moves forward while all the feet stay in contact with the ground, or a free leg moves forward. The previous paragraph addressed the issue of the placement of the foot on the free leg. With regard to the motion of the body, its necessity is determined solely by the location of c.0.g. with respect to the intersection of the Sta- bility Admitting Lines(SALs). Again at the risk of repeating ourselves, suppose the robot next wants to declare leg 3 free and move it forward but the c.0.g. is in Quad 3, as depicted in Fig. 9(b), the robot will then have to first move its body forward until the c.0.g. lies in Quad 1. In order to calculate the displacement for the body, the important factor to bear in mind is that this distance should exceed the minimum neces- sary for the new SALS to allow the next free leg to be lifted without causing instability of the robot. If the robot is allowed unhindered travel forward, a simple solution to this problem is to move the body forward by the maximum extent possible; this extent is a function of the mechanical limits of the robot.

    Stability Admitting Line 1

    7 f SubililyAdmiaingLinc2

    (C.O.G. loution)

    Figure 11: Minimum Distance the bodg needs to move forward to ensure stability during the motions to be executed by the next free leg is obtained from the geometry of the SALS.

    On the other hand, for adaptive gaits (such as when the robot must adjust its forward progress in order to either come to a stop at a particular location, or carry out collision avoidance, or veer or turn to one side or the other, etc.), one may wish to calculate a specific distance for the body motion. The min- imum distance, denoted d,, which should be exceeded by tlie motion of the body is calculated using SALs as shown in Fig. 11. Recall that we keep track of all feet positions with respect to the robot body frame. Thus, we know where feet locations of leg 1 and leg 3 are with respect to the robot body origion, which means we can compute equations for lines in Fig. 11 in the robot body coordinate frame. All that is left is finding the intersection point of SAL 2 with the y axis of the robot body coordinate frame. This is the minimal distance the body needs to move for the robot to stay stable during the next move if leg 3 is selected as the next free leg.

    An idea under development is to construct a table whose rows correspond to the different possible four-sided support patterns and whose columns correspond to the locations of the c.0.g. in terms of the Quad region occupied in Fig 6. The entries in the table will yield a particular sequence of leg and body move- ments. Such a table would be useful for the online synthesis of adaptive gaits for use during collision avoidance, etc.

    4. Conclusion

    In this paper, we introduced a new mechanical leg design for a quadruped walking robot. We proposed a new simplified straight line, periodic, monotonically forward (SLPMF) walk- ing gait for such robots. We showed that these types of gaits can be controlled by using the notion of Stability Admitting Lines. The geometry of these lines, always available to tlie controller, permits the determination of whether or not the se- lected free leg should move forward without jeopardizing the stability of the robot. And, if the stability of the robot would be violated, these lines also permit the determination of the ex- tent to which the body should move forward so as to guarantee stability during the next motion.

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