1. Introduction
In the field of humanoid research, control algorithm of walking pattern plays a significant role. There were approaches based on batch type processed to predesign the walking trajectory and/or control policy. These approaches commonly needed iterative computation to design either entire trajectories or piecewise trajectories over some time interval. As the computing power grows, these approaches became implementable and many successful results were produced [1-3]. To implement the walking pattern in real time, reduced dynamic model such as a single linear inverted pendulum was adopted [4, 5]. In addition, there were some approaches to reduce the complexity of the equation of motion by assuming that the ZMP(Zero Moment Point) trajectory has a specific form [6-9]. In several methods, explicit walking trajectories were not used. These were mainly focused on a sensory feedback algorithm with several intuitive rules [10, 11].
Improving the ability to walk according to the navigational commands is as important as improving the stability. Especially, this is important to deal with navigation within a complex environment [12, 13]. In this paper, to generate walking pattern which satisfies the complex navigational commands, such as walking period, step length and walking direction, a novel algorithm is introduced and resolves the following key points:
● Generation of Periodic Walking Patterns: Ability to generate a periodic walking pattern from the given navigational command.
● Transition between Two Walking Patterns: Ability to modify a walking pattern by the transition from current pattern to the desired one.
● Decision on Feasibility about Navigational Commands: Ability to judge whether or not a navigational command is feasible.
In order to address these subjects, analytical solutions of the equation of motion for the simplified dynamics are utilized. In addition, by adopting the ZMP variation scheme, the proposed algorithm includes the ability to modify the walking pattern without any extra step for adjusting the CM motion. Note that the proposed algorithm is focused on the feed-forward pattern generation. To make robot walk stably, it is essential to adopt some feedback controller sintroduced in [14-17].
The iteration based method such as a preview control scheme utilizes the reference ZMP trajectory as a preview data to give locally optimized walking pattern [3]. In this method, navigational command is implicitly utilized to design the reference ZMP trajectory. Since the proposed method utilizes the pre-defined ZMP functions on the convex hull within the sole, however, the walking patterns are derived from a navigational command explicitly as a closed-form [18]. Consequently, the generated walking patterns satisfy the navigational command exactly if it is feasible. Moreover, it provides a criterion which indicates the navigational command is reachable or not from the current walking state. In this paper, the previous work is further developed to handle the infeasible navigational command by introducing the binary search algorithm.
This paper is organized as follows. In Section 2, walking motion is analyzed in the view of periodic motion and transition between the different periodic motions. Desired goal state for the transition, desired walking state is derived in Section 3. The ZMP variation scheme is reviewed in Section 4. Feasibility of the navigational command is considered in Section 5. Section 6 presents example walking pattern using computer simulation. Conclusion is provided in Section 7.
2. Periodic Nature of Walking Pattern
To perform a complex navigational task, the algorithm requires a minimal command set that allows for configuration of the walking pattern. The set of parameters includes single and double supporting times, forward and side step lengths, and walking directions. This instructional set is defined as a command state.
Definition 1 Command State(CS) is a minimal set to represent navigational commands as follows:
where Tsl, Tdl, Fl, Sl and θl represent single support time, double support time, forward step length, side step length and walking direction, respectively, for the left side. Similarly, Tsr, Tdr, Fr, Sr and θr are for the right side. Note that the side step lengths in the lateral plane are for sideways movement.
Since a walking is a periodic motion, it can be represented as a limit cycle. When a certain CS is specified, the consequent walking pattern is determined. In other words, walking according to a given C can be analyzed as a movement of state along the periodic orbit which stands for the walking pattern. Therefore, in order to handle complex navigational commands, corresponding walking pattern should be figured out. Fig. 1 describes this periodic walking pattern conceptually. Dashed line and dash-dot line represent two different walking patterns, A and B conceptually. When the new navigational command,cB, is given during a robot walks according to the current navigational command, cA , new walking pattern B is designed according to the cB. Subsequently, the robot tries to trans it from the current state, which is depicted as a triangle, to the desired state on the new walking pattern B. In other words, when CS is changed into cB from cA, a corresponding walking pattern B is resolved. Subsequently, the current state travels to the desired one thereby the walking pattern is modified.
Fig. 1.Transition between two different walking patterns.
where
c(t) and s(t) are abbreviation of cosh (t/Tc) and sinh (t/Tc) with a time constant and a constant height Zc, respectively, and p(t) and q(t) represent the ZMP function on the sagittal and the lateral plane, respectively, lastly, is convolution operator. In spite of conciseness, Eq. (1) characterizes dominant dynamics as well as it separate sagittal and lateral motion.
Note that when the homogeneous solution part in Eq. (1) is only used, namely, the ZMP is assumed that it stays at the center of foot polygon, the CM motion is predetermined by given time, t . Consequently, it is unmodifiable throughout the single support phase. This equivalently means that the humanoid robot cannot accelerate or decelerate its CM. Therfore it can not vary its step length, walking period and direction in the same manner that humans do. The particular solution part, however, relaxes this restriction. By allowing a variation of ZMP over the convex hull on the bounded foot region, it is possible to change the position and the velocity of the CM independently throughout the single support phase.
For the sake of convenience, a state of walking pattern at a given time is defined as a Walking State(WS). Since in the linear inverted pendulum model, the mass of the robot is considered as a point mass, its walking motion can be represented in terms of the position and the linear velocity. Consequently, the position and the velocity of the CM become the WS as follows:
Definition 2 Walking State (WS) for the linear inverted pendulum represents the CM state as follows:
Note that the velocity term is weighted with the time constant, Tc. From now on, walking motion is analyzed in terms of the WS.
3. Desired Walking State
Modification of a walking pattern can be characterized into two parts: generation of another periodic walking pattern which satisfies a given new CS and transition of WS from a current state to a desired state on the new trajectory.
Fig. 3 represents two different walking patterns and transitions between them, where the limit cycle means a new walking pattern which is generated by the new CS. The triangle and the circle represent current and desired WS, respectively. Lastly, solid line from the triangle to the circle indicates transitional walking pattern. As shown in Fig. 3 (a), transition from the current state can be accomplished in various ways. In other words, a lot of desired WS and transitional walking patterns are possible. Therefore it arise an optimization issue. In practical manner, desired WS is restricted to a final state of single support phase which is depicted as a circle on a solid-line for the single support phase. While a dashed-line means double support phase on walking pattern B as shown in Fig. 3 (b). Moreover, it is assumed that the ZMP variation carries out only during the transition. Consequently, WS on the limit cycle is characterized by only the homogeneous solution part of Eq. (1). Using this strategy, the desired WS is uniquely determined from a given CS and a current WS as shown in Fig. 4 which describes a periodic walking pattern according to the CS.
Fig. 3.Transitions of WS. (a) Various possible transitions. (b) Proposed transition strategy
Fig. 4.Foot print of steady walking.
In the figure, and subscripts, li(f) and ri(f) , mean left initial(final) and right initial(final) state of the single support, respectively. Lastly, θ implies walking direction.From Eq. (1), motions of the single support phase are derived as follows:
As mentioned above, during the periodic motion ZMP does not vary and stays at the center of foot polygon. Consequently, the motion of equation is characterized with only homogeneous part. Also, double support motions which are controlled with constant velocity are obtained as follows:
where
By eliminating the initial WS from Eq. (2) and (3), homogeneous solutions of the steady motion are given as follows:
Using the Kronecker Product, Eq. (4) is transformed into the familiar linear equation form as follows:
where
Finally, solving Eq. (5) for the final WSs, 𝛇lf and 𝛇rf, desired WSs are given as follows:
Eq. (6) expresses the mapping relationship between the CS and the desired WS. Note that the information of the CS is involved in the matrix A* and B*. Whenever the CS is changed, it translates into the desired WS form. Subsequently, 𝛇lf or 𝛇rf becomes the desired WS for the left or right support phase, respectively.
4. ZMP Variation Scheme
After the desired WS is calculated, it is necessary to adjust the ZMP trajectory to transfer the current WS to the desired one. Via the variation of the ZMP with closed-form function, it is possible to modify the walking pattern as intended by the CS. In a practical manner, a constant function(Fig. 5 (a)) and a step function(Fig. 5 (b)) are proposed for sagittal motion and lateral motion, respectively. Detailed analysis of these two ZMP functions is covered in the previous work [18].
Fig. 5.Proposed ZMP functions. (a) Constant function for the sagittal motion. (b) Step function for the lateral motion.
The control parameters ( T, P, Tsw and Q ) which characterize the proposed ZMP functions are now solved to ensure that the desired WS is attainable.
4.1 Sagittal motion
By substituting t = T and p(t) = P into Eq. (1), following solutions are obtained:
where the subscripts c and d and represent current and desired state, respectively. Note that T represents the requiredtransition time from the current WS to the desired WS.
4.2 Lateral motion
Since the homogeneous part of Eq. (1) for the lateral motion is determined by the current WS and the transition time T , it is only necessary to consider the particular part. Similarly to the sagittal motion, by letting
the particular solution of the lateral equation of motion is derived. Solving this for the unknowns, Tsw and Q, gives
where
Using Eq. (7) and Eq. (8), control parameters are directly obtained from the current and the desired WS. Subsequently, WS is transited by the ZMP variation.
5. Modification of Infeasible Command State and Desired Walking State
As explained in the previous section, a desired WS is achieved by the ZMP variation. Due to the several kinematical and/or dynamical constraints, however, sometimes desired WS is not attainable. For instance, when the magnitude of the ZMP function for the sagittal and/or lateral motion is larger or smaller than the maximum or the minimum limit, there is no way to achieve the desired WS. In this case, the WS should be substituted to the nearest one.This is illustrated in Fig. 6, where the empty circle means an infeasible desired WS whereas the solid circle means a feasible desired WS substituted from the original one.
Fig. 6.Modification of desired WS corresponding to infeasible CS. (a) Infeasibility of desired WS. (b) New walking pattern from the modified CS
Since the substituted WS is not on the desired walking pattern B (see Fig. 6 (a).), it is necessary to re-compute a new walking pattern B* through the new WS (see Fig. 6 (b).). To do so, however, modification of the original CS has to be preceded because the walking pattern is dependent on the CS. Subsequently, the current WS can travel to the replaced feasible WS on the new walking pattern.
In this paper, infeasible CS and desired WS are simultaneously modified by using Binary Search Algorithm as shown in Algorithm 1, where subscripts p and d represent previous and desired state, respectively, and m and f represent modified and feasible state, respectively, lastly, f(·) means a function gives a desired WS from a CS by Eq. (6). After initialization, CS is updated as a mean value between previous and desired CS. Note that the modified CS is always feasible because the current WS is on the walking pattern generated by the previous CS. Subsequently, if the modified desired WS is feasible then the modified desired WS moves to the desired one, else it moves to the previous one until the iteration reaches to the maximum. This Binary Search Algorithm is simple and intuitive. Moreover it is sufficiently fast to execute in real time.
6. Example Walking Pattern
The proposed algorithm was simulated to verify the effectiveness with several example walking patterns including forward walking, backward walking, sideways walking, and turning motion. Parameters used in simulation are shown in Table 1, where Zc means constant CM height in the linear inverted pendulum model (see Fig. 2.) and, Pmax(min) and Qmax(min) represent maximum(minimum) ZMP variation region for sagittal and lateral motion, respectively. Note that all length units are given in meters.
Table 1.Simulation parameters.
Fig. 2.Linear inverted pendulum model.
6.1 Forward walking
Forward walking patterns were realized from the following CS list (see the Definition 1):
Initial CS, c = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 − 0.2 0.0° 0.0°]T After the 2nd step, c = [0.3 0.3 0.05 0.05 0.4 0.4 0.2 − 0.2 0.0° 0.0°]T After the 5th step, c = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 − 0.2 0.0° 0.0°]T After the 6th step, c = [0.4 0.4 0.1 0.1 0.0 0.0 0.2 − 0.2 0.0° 0.0°]T
where time and length units are given in seconds and meters, respectively, and angle unit is given in degree. Note that the CS after the kth step is not delivered until the double support phase of the kth step is finished. In other words, the proposed algorithm does not use preview data. In the all example walking patterns during this simulation, robot was commanded to stop after accomplishing the final CS.
Fig. 7 shows the walking pattern generated by the proposed method. The solid curve and the rectangle indicate the CM trajectory and the foot polygon, respectively. The circle and the number mean the center of foot and the step number, respectively. The initial foot positions are (0, 0) for the right foot and (0, 0.2) for the left foot, respectively. As the CS list intended, robot lengthened its stride after the 2nd step and shortened after the 5th step. Although the robot was commanded to lengthen its step length as 0.4m after the2nd step, however, the step length is increased smoothly for 0.3108(= 0.7108 — 0.4) m at the 3rd step and 0.3792(= 1.09 — 0.7108) m at the 4th step, respectively. It finally achieved the commanded step length, 0.4(= 1.49 — 1.09) m, at the 5th step. Also it can be seen that similar result is observed at the 6th step. These were caused by that the infeasible CSs delivered after the2nd and the 5th steps were substituted for the relaxed CS which is thereby feasible.
Fig. 7.Example pattern (forward walking).
6.2 Backward walking
Backward walking patterns were realized from the following CS list:
Initial CS, c = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 −0.2 0.0° 0.0°]T After the 2nd step, c = [0.4 0.4 0.1 0.1 −0.3 −0.3 0.2 −0.2 0.0° 0.0°]T After the 4th step, c = [0.4 0.4 0.1 0.1 −0.1 −0.1 0.2 −0.2 0.0° 0.0°]T After the 5th step, c = [0.4 0.4 0.1 0.1 0.0 0.0 0.2 −0.2 0.0° 0.0°]T
Fig. 8 shows the backward walking pattern generated from the CS list. According to the CS list, walking patterns were generated to walk backward and stop after the 5th step. Similar with the forward walking case, although the commanded backward step length was 0.3 m after the 2nd step, it is not satisfied until the 4th step. It is finally achieved for the step length at the 4th step.
Fig. 8.Example pattern (backward walking).
6.3 Sideways walking
Sideways walking patterns were realized from the following CS list:
Initial CS, c = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 −0.2 0.0°0.0°]T After the 2nd step, c = [0.4 0.4 0.1 0.1 0.1 0.1 0.2 −0.4 0.0° 0.0°]T After the 5th step, c = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 −0.2 0.0° 0.0°]T After the 6th step, c = [0.4 0.4 0.1 0.1 0.0 0.0 0.2 −0.2 0.0° 0.0°]T
Fig. 9 shows the sideways walking pattern. According to the CS list, side step lengths are designed as different as 0.2 m and −0.4 m for the left and the right step, respectively. Via the 3rd step, which has a small difference between the commanded and the generated side step lengths due to the relaxation of the infeasible CS, subsequent footsteps were followed the given CSs accurately.
Fig. 9.Example pattern (sidewayswalking).
6.4 Turning motion
Turning motions were realized from the following CS list:
Initial CS, = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 −0.2 0.0° 0.0°]T After the 2nd step, = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 −0.2 0.0° −60.0°]T After the 5th step, = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 −0.2 0.0° 0.0°]T After the 6th step, = [0.4 0.4 0.1 0.1 0.0 0.0 0.2 −0.2 0.0° 0.0°]T
Fig. 10 shows the turning motion. After the 2nd step, it started to turn 30° to the right and stopped after the6th step.
Fig. 10.Example walking (turning motion).
6.5 Complex walking
Complex walking patterns including forward, backward, sideways and turning motion were realized from the following CS list:
Initial CS, c = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 − 0.2 0.0°0.0°]T After the 2nd step, c = [0.4 0.4 0.1 0.1 0.2 0.2 0.2 − 0.2 0.0° − 30.0°]T After the 4nd step, c = [0.4 0.4 0.1 0.1 − 0.2 − 0.2 0.4 − 0.2 0.0° − 30.0°]T After the 7th step, c = [0.4 0.4 0.1 0.1 − 0.2 − 0.2 0.2 − 0.2 0.0° 0.0°]T After the 8th step, c = [0.4 0.4 0.1 0.1 0.0 0.0 0.2 − 0.2 0.0° 0.0°]T
Fig. 11 shows the generated walking pattern. After the 2nd step, the wakling direction was changed to the right. After the 4th step, it started to walk backward with different side step length. Subsequently, it stopped after the 8th step.
Fig. 11.Example pattern (complex walking).
7. Conclusion
The modifiable walking pattern generation algorithm was proposed. By varying the ZMP(Zero Moment Point) while in a single support phase, it is possible to move the current walking state, the position and the velocity of the CM(Center of Mass), to the desired one. Consequently, the humanoid robot could change its step length, walking period and direction, independently. In addition, when the navigational command is infeasible, it is substituted for the relaxed one by the binary search algorithm and thereby becomes feasible. As a result, walking pattern could be successfully generated even though the command was infeasible. Since the proposed method utilizes the closedform functions and simple search algorithm to generate walking pattern, it is possible to compute in real time.
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