Stability and controllability in a rising motion - Semantic Scholar
Stability and controllability in a rising motion: a global dynamics approach Tomoyuki Yamamoto∗ and Yasuo Kuniyoshi† Humanoid Interaction Lab. Electrotechnical Laboratory National Institute of Advanced Industrial Science and Technology (AIST) C2, 1-1-1 Umezono, Tsukuba, 305-8568 Japan E-mail: [email protected], [email protected]
Abstract A novel concept for controlling the humanoid robot, the global dynamics is investigated by motion capture experiments. This concept to generalise human/humanoid body motion as a transition of “envelopes”, where body dynamics is exploited and control input is adopted only at the “nodes”, where the body is unstable and control input is necessary. For the experiment, a dynamical rising is chosen and full body motion is measured. By evaluating coordination between joint angles, we have seen variation of motion corresponds volume of envelope (stable region within the phase space). Also, by analysis in the phase space (i.e., include variables both of positions and their time derivatives), we have seen variation not only according to boundary condition for the motion (i.e., adopting physical restriction) but also according to experience. Our results shows that the body dynamics is so complicated that even single type of motion may have variation. This also means controllability of human(oid) motion in dynamically stable region. We believe these results contributes for thinking design of flexible motion, as the concept of the global dynamics suggests.
1
Introduction
As humanoid robots obtain human-like versatility in mechanical sense, it seems to be a hard problem to design their behaviors. While the human do not seems to suffer a problem to generate its behaviors on demand in everyday-life situation, for humanoid robots it is a very hard problem. Most commonly ∗ recent address: School of Knowledge Science, JAIST, Ishikawa 923-1292 Japan † recent address: Dept. of Mechano Informatics, Univ. of Tokyo, 113-8654 Japan
used strategy for trajectory generation, the high-gain servo method (e.g., used by [1]) is basically playing back of the predefined ideal trajectory. Then if we expect a robot to behave as various as the human, almost infinite number of ideal trajectory may be required. We argue that this is not a desired situation, hence an alternative approach, the global dynamics[2][3] is proposed. Our idea is basically to exploit body dynamics and minimise control input. Although body dynamics is nonlinear and looks complicated, if we investigate its phase space, there are stable and unstable regions coexists. Here, stability can be divided into three parts. Aside trivial static stability, the rest of two should be described. One is the ZMP-guaranteed (ZMP: Zero Moment Point [4]) stability, where the body is guaranteed that it does not fall. As a special case of this stability, when the velocity of CG (Center of Gravity) is small, we call it “kinematically guaranteed stability”. Because in such case, the contribution of total momentum for ZMP is small and position of CG is most important factor. The position of CG is defined by angle of each joint and the stability can be maintained by adopting boundary conditions for the joints and the stability is kinematically achieved. The other is passive stability, where stability is maintained by interaction between dynamical modes, as is done by passive dynamic walking by McGeer[5]. In this famous example, around the support transfer phase of a passive walking machine, he found that three modes (i.e., speed, swing, and totter) interact each other to form oscillatory relationship that has eigenvalues of complex number. This relationship has a role of a buffer that absorbs perturbation and stability is dynamically maintained. This idea is already adopted for specific movements, such as walking ([5][6][7]), we are trying to estab-
node envelope
Figure 1: A schematic representation of the global dynamics. Several ways of rising are shown. For each snapshots, a schematic instantaneous landscape of potential energy is shown. Gray regions are envelopes and dotted circles are nodes (see text for detail).
lish a generic method for human/humanoid behavior. Suppose that we have good “map” of the phase space of body dynamics, our idea is to exploit stable region and apply control input around exchanging (i.e., unstable) point between stable regions. Then unstable regions can be regarded as nodes for switching motion and we expect a behavior is described in ”node-to-node” manner, as a sequence of stable motions. Here it is important to note that “nodeto-node” means the control input is now treated as discreet time and we expect to describe control in a state machine-like manner. To establish this concept, we are studying in two directions. One is dynamical simulation of simple body [3] where useful several features of elastic body are shown. The other is experimental study of human body, which is described in the present work. In the former, we have shown that an elastic body has several merits of exploiting passive stability, oscillation damping and a whip-like motion. In the latter, as will be shown below, we could see small scale variation within a single type of motion that may imply high dimensional structure of the phase space is exploited. We think this feature strongly support the relevance of the global dynamics and helps to detail investigation of it.
2
The Global Dynamics
Let us describe the global dynamics in more detail. In fig. 1, a schematic picture is shown. This figure describes possible patterns of rising motion. Several patterns of rising motions are shown, which corre-
sponds variety of rising pattern of even single person when no description is given, as is reported by van Sant [8][9]. Each pattern includes snapshots, where posture and schematic potential energy landscape (as snapshots) are shown. Gray areas are stable regions, which are called “envelopes”. Within each envelope, the trajectory is stable by passive (i.e. dynamic) or static (i.e. trivial or kinematically guaranteed) stability. In these regions, the body can be run by predefined parameter and the computation cost for the control is minimised. While the control is easy within each envelope, transition between envelopes may necessary along the trajectory of a motion. Between envelopes, unstable regions exist as is shown as circles bordered by a dotted line in the figure. In such regions, while the body is unstable, it is very important to note that we can exploit the instability for switching between envelopes. We call these unstable regions the “nodes” (dotted circles in the figure). The node interconnects envelopes and can be regarded as controlling points. Because of its instability, the transition at the node can be done by adding small perturbation, which we call “intervention”. Then we think the motion design can be drastically simplified. Suppose that we have a map of stable/unstable regions, the desired path can be approximated as a succession of envelopes and nodes. As mentioned earlier, the computation cost of control is minimised within an envelope and higher level controller should be active only at the nodes to chose an envelope as the next state (i.e., envelope). From a different angle, the nodes segments the dynamics and if it is possible to sense its instablity in real time manner, the nodes may tell when to add control. Although we do not investigate the nodes in the present work. While we have studied the dynamics of a simple body [3], study of the whole body, which consists of more than 10 segments, is a hard problem. Then an experimental study of real human body is essential.
3
Experiment
We have carried out motion capture experiments of a rising motion. By investigating whole body motion, we expect dynamical aspects of the global dynamics are revealed. As for materials, we use ExpertVision system by Motion Analysis Corp. (CA, USA) with 10 cameras. We have carried out motion capture experiment of the rising motion. So far five healthy subjects are participated. To analyse full body motion, we have adopted the ETL/HH marker set[10] which enables full body analysis of 19 segments and 16 joint angles.
In this marker set, the torso is divided into three segments and flection/extraction of the spine can be evaluated. The motion pattern that we take for the experiment is a dynamical rising, that swing up the legs and rise the body without using hands, as is shown shadowed envelope on fig.1. The motion can be divided into three parts, the swing phase, the contact phase and the standing up phase. This pattern requires angular momentum around the support transfer phase (i.e., when supporting point is moved from hip to foot) and dynamical feature is expected to be found. There is a variation in the experiment. We adopt an obstacle between the hip and the foot, so that support transfer must require nonzero angular momentum. We expect this restriction may show a diversification of the trajectories.
4
Result
So far we have analysed one subject and the rest is being processed. Due to high density of markers and frequent occulution of markers, the identifying process requires much more work than other motion, such as walking. Since the phase space of the whole body of 19segment system is about 30 dimentional and it cannot be analysed as a whole. Then we have extracted as a coordination between variables, as a projection. Then we have evaluated the results in two aspects. One is the coordination between joints, where only dimension of angle (i.e., position) is considered. The other includes a term of time derivative (i.e., speed) and more dynamical aspect can be investigated. 4.1
coordination between joints
First, let us discuss relationship between joints. In fig.2 angular relationship between the left hip and the left knee is shown. Data from 11 trials are superimposed. The degree of coordination can be roughly evaluated by convergence of the plots. In the figure, when the legs are swung up, the plots do not converge while the trajectories are similar. We think this variation comes from that the human reduces cost of precise control for this region of the motion, since in this posture, the body is supported by the back and the leg position does not contributes to the stability of the whole body. In other words, in this region, the volume of envelope is big. On the other hand, when the body is standing up, trajectories rather converge. In this phase, the hip angle greatly contributes to the attitude of the upper body. In this phase, the stability is achieved by
kinematically guaranteed stability and COP must be within the foot print. Then strict coordination between the hip and the knee is expected. It is important to note that the trajectories do not completely converge: there is still a region of stability. This feature implies that the volume of the envelope is small in this phase. Between above two phases, the contact phase exists. At this phase, although the plots are missing in in many trials due to occulution of the markers, we expect the plots are diverging. In this phase, the supporting surface is changed from the hip to the foot. Because of the change of the boundary condition, the direction of the body is changed to upward and this direction is unstable, while other directions are stable. Then this phase can be regarded as a saddle point, where the body is unstable and the trajectory strongly depends on the perturbation. It is very important to note that instability and strong dependence on the perturbation also mean the controllability. Because small perturbation induces large difference in short period of time, it can be regarded as an efficient way of controlling. In the phase space, there are many such saddle points or unstable regions. If they connect several envelopes, we call them “nodes”, as control points. Also, the perturbation adopted at the nodes is called “intervention”. Although the method for the intervention is being studied, as suggested in [3], an indirect way, like sudden change of parameter is regarded as a good candidate. Recall that in the passive dynamic walking, interaction between modes around the support transfer phase (i.e., a node in our theory) is essential. Then direct intervention may void the interaction and destroy its stability. 4.2
analysis of phase space by evaluating speed
While so far only angular information is discussed, secondly, let us investigate dynamical aspect of rising more in depth by including momentum. We found that the difference between trials of two different descriptions becomes clear when momentum variable is considered. In fact, in fig. 2, trials of two different descriptions (i.e., regardless of whether restriction for foot placement is adopted or not) are superimposed and obvious difference is not seen. However, if we choose different pair of variables, the difference become clear. Between fig.3 (without restriction) and fig.4(with restriction on foot placement), two plots are topologically different. In these figures, several plots of the relation between head speed (i.e. norm of velocity) and angle of the ankle are superimposed. The head speed corresponds
to angular momentum of the upper body. We may retard this pair of variables roughly shows the coordination between lower and upper body. As mentioned, when foot placement is restricted to forward, then when foot placement is restricted forward, larger angular momentum is required to achieve support transfer (i.e., from the hip to the foot) at the contact phase. This feature is seen when we compare around the upper left of the plots of figs. 3 and 4, where crossing of the plot is seen in the latter.
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This crossing region corresponds later half of the swing phase: i.e., swing forward the whole body to gain angular momentum. The steep plot of fig. 4 implies that higher speed is obtained, in preparation for the support transfer, where larger angular momentum is required. Also, in the support transfer phase, faster speed is shown in fig. 4 (around the right loop), for the same reason as above.
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Although the number of data is a few, the plots looks diverge around the swing phase. This feature agrees that the degree of convergence of trajectory corresponds the volume of envelope, as is stated above.
Figure 2: superimposed plot of relationship between angles of left hip and left ankle
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Finally, we should mention that while this result is obtained from experiment of single subject and needs more data for further study, the data currently being processed seems to support the result obtained here.
variation exists even within the same motion
Finally, we describe a differentiation within same motion. We found that in some of trials (of without restriction condition) shows different pattern of trajectory, when momentum variable is considered. In fig. 5, a super imposed plot of the relationship between the head speed and the ankle is shown. This plot looks like fig. 3: although they are topologically same, the shapes are clearly different. In fact, what make divide these two cases is the number of trials. In the fig. 5, two last trial of ten trials are plotted and first eight trials are shown in fig. 3. This difference is not expected, since we did not recognise this small (but clear) difference at the experiment. Watching the animation of captured data, even in it is not easy to tell the difference. As an impression, in the latter case (i.e., fig5) the motion becomes smooth: we assume it is because of an “optimisation” process of a human through experience. Since higher speed is achieved around the support transfer in fig 5, the motion is smooth and regarded as efficient. Then we call the first eight trials as “unexperienced” rising and the last two trials as “experienced” rising. In this experiment it is shown that trajectory is variable even within an envelope. This feature is very important when one think about design of humanoid motion, since this variation may leads to the controllability within an envelope.
5
Concluding Remarks
In the present work, through experimental study of a rising motion, we have revealed complex structure of the phase space of a rising motion and its relation to the global dynamics. Then we have shown that the degree of convergence of a bundle of trajectory corresponds the volume of envelope. Although this is expected, even within an envelope, variation is seen. This “microscopic” variation is unexpected. Since the envelope is regarded as a stable region, we expected convergence for all trials with small change as a result of experience. However, we have seen obvious differentiation. We think this differentiation may be due to localisation within an envelope. Since an envelope occupies a part of the phase space as a volume and its dimension is as high as that of the phase space. Then there are variation of trajectories for the same type of motion. Note that plots shown in this paper are two-dimensional and they are projection of original high dimensional manifold. Then study of a good method and choice of variables for projection is suggested as a future plan. Also, we need to a method to evaluate similarity between different pattern of coordinations. While
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we could distinguish several patterns, as general structure of nonlinear dynamics suggests [11], even smaller scale of microscopic structure may exist. We should note that by exploiting such microscopic structure, we may obtain controllability within an envelope to realise human-like flexible behavior.
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It is important to note that we do not focus on searching for an ideal trajectory based on some kind of optimisation. Human body is not a precision device and it is wise to chose a trajectory where stability is guaranteed by its own dynamics, such as within an envelope. A humanoid robot can be a precision device but the environment is not. Then it seems to be always a good idea to exploit a stable dynamical structure, whenever applicable. If we think about the relevance of our results for humanoid robotics, it is important to note that the dynamics discussed in the paper is that of a human’s body. The dynamics of human body includes muscle as elastic elements, which is not usually included in the robot’s mechanism. One possible difference is that volume of the envelope may be enlarged, as is shown in our former work [3]. To exclude this effect, we are going to feed the obtained trajectory to a dynamics simulator. However, we expect the essence, complex network of envelopes and nodes, is preserved. While so far we have discussed on the envelope, the node must be analysed in the future. Because experimental analysis of unstable structure requires much more data than that of stable structure and we could not investigate in this work. As mentioned above, numerical analysis by a dynamics simulator may be helpful. As for experiment, while we are processing the captured data, for the rising, this is much harder than other motion, because of frequent occulution of markers, quick rotation of body segments and high density of markers. This problem is expected to be solved by improvement of algorithm of motion capture system. In near future, we are going to establish the notion obtained here by collecting many data.
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Although detail mapping of the phase space is required, we think that by choosing “microscopic nodes”, fine tuning of behavior is possible. Then, the design of motion based on the global dynamics will be done in two stages: first choose rough trajectories by choosing nodes (i.e., macroscopic) and then tune the behavior by choosing microscopic nodes.
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ankle [deg] Figure 5: superimposed plot of relationship between head speed and angle of left ankle. No restriction is adopted for the foot placement. Two last trials of ten trials are shown.
To sum up, through the motion capture experiments of a human rising motion, we have shown complicated structure of the phase space. The degree of convergence within a bundle of trajectories corresponds the area of the envelope and this supports our
concept of the global dynamics. Also, investigation of the phase space suggests that microscopic structure exists, even in single envelope. By analysing and exploiting, it may lead to a flexible motion of a humanoid robot. acknowledgment The authors are greatly thankful for Drs. G. Taga, K. Kaneko for discussions. Also the authors are thankful for Mr Y. Kawamura and T. Nagasaki for assisting experiments. The concept of global dynamics originally suggested by Dr. A. Nagakubo and one of the authors (YK). This work is supported by COE program funded by STA (Science and Technology Agency of Japan).
References [1] K. Hirai, M. Hirose, Y. Haikawa, and T. Takenaka. The development of honda humanoid robot. IEEE International Conference on Robotics and Automation, 1998. [2] Yasuo Kuniyoshi and Akihiko Nagakubo. Humanoid as a research vehicle into flexible complex interaction. IEEE International Conference on Intelligent Robots and Systems, pages 811–819, 1997. [3] Tomoyuki Yamamoto and Yasuo Kuniyoshi. Harnessing the robot’s body dynamics: a grobal dynamics apporach. IEEE International Conference on Intelligent Robots and Systems, pages 518–525, 2001. [4] M. Vukobratovic and J. Stepanenko. On the stability of anthropomorphic systems. Mathematical Biosciences, 15:1–37, 1972. [5] Tad McGeer. Passive dynamic walking. Int. J. of Robotic Res., 9(2):62 – 82, 1990. [6] Tad McGeer. Passive walking with knees. Proc. IEEE Conf. on Robotics & Automation, pages 1640 – 1645, 1990. [7] J. Pratt and G. Pratt. Exploiting natural dynamics in the control of a planar bipedal walking robot. In Progeedings of the Thirty-Sixth Annual Allerton Conference on Communication, Control, and Computing, pages 739–748. 1998. [8] A. van Sant. Rising from a supuine position to erect stance. Physical Therapy, 68:185–193, 1988. [9] A. van Sant. Age differences in movement patterns used by children to rise from a supuine position to erect stance. Physical Therapy, 68:1330–1339, 1989. [10] Tomoyuki Yamamoto and Yasuo Kuniyoshi. Introduction of the etl/hh marker set for whole body motion analysis (in Japanese). proceedings of 19th annual conference of RSJ, pages 597–598, 2001. [11] V. I. Arnold and A. Avez. Ergodic Probrems of Classical Mechanics. W A Benjamin, 1968.
Abstract A novel concept for controlling the humanoid robot, the global dynamics is investigated by motion capture experiments. This concept to generalise human/humanoid body motion as a transition of “envelopes”, where body dynamics is exploited and control input is adopted only at the “nodes”, where the body is unstable and control input is necessary. For the experiment, a dynamical rising is chosen and full body motion is measured. By evaluating coordination between joint angles, we have seen variation of motion corresponds volume of envelope (stable region within the phase space). Also, by analysis in the phase space (i.e., include variables both of positions and their time derivatives), we have seen variation not only according to boundary condition for the motion (i.e., adopting physical restriction) but also according to experience. Our results shows that the body dynamics is so complicated that even single type of motion may have variation. This also means controllability of human(oid) motion in dynamically stable region. We believe these results contributes for thinking design of flexible motion, as the concept of the global dynamics suggests.
1
Introduction
As humanoid robots obtain human-like versatility in mechanical sense, it seems to be a hard problem to design their behaviors. While the human do not seems to suffer a problem to generate its behaviors on demand in everyday-life situation, for humanoid robots it is a very hard problem. Most commonly ∗ recent address: School of Knowledge Science, JAIST, Ishikawa 923-1292 Japan † recent address: Dept. of Mechano Informatics, Univ. of Tokyo, 113-8654 Japan
used strategy for trajectory generation, the high-gain servo method (e.g., used by [1]) is basically playing back of the predefined ideal trajectory. Then if we expect a robot to behave as various as the human, almost infinite number of ideal trajectory may be required. We argue that this is not a desired situation, hence an alternative approach, the global dynamics[2][3] is proposed. Our idea is basically to exploit body dynamics and minimise control input. Although body dynamics is nonlinear and looks complicated, if we investigate its phase space, there are stable and unstable regions coexists. Here, stability can be divided into three parts. Aside trivial static stability, the rest of two should be described. One is the ZMP-guaranteed (ZMP: Zero Moment Point [4]) stability, where the body is guaranteed that it does not fall. As a special case of this stability, when the velocity of CG (Center of Gravity) is small, we call it “kinematically guaranteed stability”. Because in such case, the contribution of total momentum for ZMP is small and position of CG is most important factor. The position of CG is defined by angle of each joint and the stability can be maintained by adopting boundary conditions for the joints and the stability is kinematically achieved. The other is passive stability, where stability is maintained by interaction between dynamical modes, as is done by passive dynamic walking by McGeer[5]. In this famous example, around the support transfer phase of a passive walking machine, he found that three modes (i.e., speed, swing, and totter) interact each other to form oscillatory relationship that has eigenvalues of complex number. This relationship has a role of a buffer that absorbs perturbation and stability is dynamically maintained. This idea is already adopted for specific movements, such as walking ([5][6][7]), we are trying to estab-
node envelope
Figure 1: A schematic representation of the global dynamics. Several ways of rising are shown. For each snapshots, a schematic instantaneous landscape of potential energy is shown. Gray regions are envelopes and dotted circles are nodes (see text for detail).
lish a generic method for human/humanoid behavior. Suppose that we have good “map” of the phase space of body dynamics, our idea is to exploit stable region and apply control input around exchanging (i.e., unstable) point between stable regions. Then unstable regions can be regarded as nodes for switching motion and we expect a behavior is described in ”node-to-node” manner, as a sequence of stable motions. Here it is important to note that “nodeto-node” means the control input is now treated as discreet time and we expect to describe control in a state machine-like manner. To establish this concept, we are studying in two directions. One is dynamical simulation of simple body [3] where useful several features of elastic body are shown. The other is experimental study of human body, which is described in the present work. In the former, we have shown that an elastic body has several merits of exploiting passive stability, oscillation damping and a whip-like motion. In the latter, as will be shown below, we could see small scale variation within a single type of motion that may imply high dimensional structure of the phase space is exploited. We think this feature strongly support the relevance of the global dynamics and helps to detail investigation of it.
2
The Global Dynamics
Let us describe the global dynamics in more detail. In fig. 1, a schematic picture is shown. This figure describes possible patterns of rising motion. Several patterns of rising motions are shown, which corre-
sponds variety of rising pattern of even single person when no description is given, as is reported by van Sant [8][9]. Each pattern includes snapshots, where posture and schematic potential energy landscape (as snapshots) are shown. Gray areas are stable regions, which are called “envelopes”. Within each envelope, the trajectory is stable by passive (i.e. dynamic) or static (i.e. trivial or kinematically guaranteed) stability. In these regions, the body can be run by predefined parameter and the computation cost for the control is minimised. While the control is easy within each envelope, transition between envelopes may necessary along the trajectory of a motion. Between envelopes, unstable regions exist as is shown as circles bordered by a dotted line in the figure. In such regions, while the body is unstable, it is very important to note that we can exploit the instability for switching between envelopes. We call these unstable regions the “nodes” (dotted circles in the figure). The node interconnects envelopes and can be regarded as controlling points. Because of its instability, the transition at the node can be done by adding small perturbation, which we call “intervention”. Then we think the motion design can be drastically simplified. Suppose that we have a map of stable/unstable regions, the desired path can be approximated as a succession of envelopes and nodes. As mentioned earlier, the computation cost of control is minimised within an envelope and higher level controller should be active only at the nodes to chose an envelope as the next state (i.e., envelope). From a different angle, the nodes segments the dynamics and if it is possible to sense its instablity in real time manner, the nodes may tell when to add control. Although we do not investigate the nodes in the present work. While we have studied the dynamics of a simple body [3], study of the whole body, which consists of more than 10 segments, is a hard problem. Then an experimental study of real human body is essential.
3
Experiment
We have carried out motion capture experiments of a rising motion. By investigating whole body motion, we expect dynamical aspects of the global dynamics are revealed. As for materials, we use ExpertVision system by Motion Analysis Corp. (CA, USA) with 10 cameras. We have carried out motion capture experiment of the rising motion. So far five healthy subjects are participated. To analyse full body motion, we have adopted the ETL/HH marker set[10] which enables full body analysis of 19 segments and 16 joint angles.
In this marker set, the torso is divided into three segments and flection/extraction of the spine can be evaluated. The motion pattern that we take for the experiment is a dynamical rising, that swing up the legs and rise the body without using hands, as is shown shadowed envelope on fig.1. The motion can be divided into three parts, the swing phase, the contact phase and the standing up phase. This pattern requires angular momentum around the support transfer phase (i.e., when supporting point is moved from hip to foot) and dynamical feature is expected to be found. There is a variation in the experiment. We adopt an obstacle between the hip and the foot, so that support transfer must require nonzero angular momentum. We expect this restriction may show a diversification of the trajectories.
4
Result
So far we have analysed one subject and the rest is being processed. Due to high density of markers and frequent occulution of markers, the identifying process requires much more work than other motion, such as walking. Since the phase space of the whole body of 19segment system is about 30 dimentional and it cannot be analysed as a whole. Then we have extracted as a coordination between variables, as a projection. Then we have evaluated the results in two aspects. One is the coordination between joints, where only dimension of angle (i.e., position) is considered. The other includes a term of time derivative (i.e., speed) and more dynamical aspect can be investigated. 4.1
coordination between joints
First, let us discuss relationship between joints. In fig.2 angular relationship between the left hip and the left knee is shown. Data from 11 trials are superimposed. The degree of coordination can be roughly evaluated by convergence of the plots. In the figure, when the legs are swung up, the plots do not converge while the trajectories are similar. We think this variation comes from that the human reduces cost of precise control for this region of the motion, since in this posture, the body is supported by the back and the leg position does not contributes to the stability of the whole body. In other words, in this region, the volume of envelope is big. On the other hand, when the body is standing up, trajectories rather converge. In this phase, the hip angle greatly contributes to the attitude of the upper body. In this phase, the stability is achieved by
kinematically guaranteed stability and COP must be within the foot print. Then strict coordination between the hip and the knee is expected. It is important to note that the trajectories do not completely converge: there is still a region of stability. This feature implies that the volume of the envelope is small in this phase. Between above two phases, the contact phase exists. At this phase, although the plots are missing in in many trials due to occulution of the markers, we expect the plots are diverging. In this phase, the supporting surface is changed from the hip to the foot. Because of the change of the boundary condition, the direction of the body is changed to upward and this direction is unstable, while other directions are stable. Then this phase can be regarded as a saddle point, where the body is unstable and the trajectory strongly depends on the perturbation. It is very important to note that instability and strong dependence on the perturbation also mean the controllability. Because small perturbation induces large difference in short period of time, it can be regarded as an efficient way of controlling. In the phase space, there are many such saddle points or unstable regions. If they connect several envelopes, we call them “nodes”, as control points. Also, the perturbation adopted at the nodes is called “intervention”. Although the method for the intervention is being studied, as suggested in [3], an indirect way, like sudden change of parameter is regarded as a good candidate. Recall that in the passive dynamic walking, interaction between modes around the support transfer phase (i.e., a node in our theory) is essential. Then direct intervention may void the interaction and destroy its stability. 4.2
analysis of phase space by evaluating speed
While so far only angular information is discussed, secondly, let us investigate dynamical aspect of rising more in depth by including momentum. We found that the difference between trials of two different descriptions becomes clear when momentum variable is considered. In fact, in fig. 2, trials of two different descriptions (i.e., regardless of whether restriction for foot placement is adopted or not) are superimposed and obvious difference is not seen. However, if we choose different pair of variables, the difference become clear. Between fig.3 (without restriction) and fig.4(with restriction on foot placement), two plots are topologically different. In these figures, several plots of the relation between head speed (i.e. norm of velocity) and angle of the ankle are superimposed. The head speed corresponds
to angular momentum of the upper body. We may retard this pair of variables roughly shows the coordination between lower and upper body. As mentioned, when foot placement is restricted to forward, then when foot placement is restricted forward, larger angular momentum is required to achieve support transfer (i.e., from the hip to the foot) at the contact phase. This feature is seen when we compare around the upper left of the plots of figs. 3 and 4, where crossing of the plot is seen in the latter.
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This crossing region corresponds later half of the swing phase: i.e., swing forward the whole body to gain angular momentum. The steep plot of fig. 4 implies that higher speed is obtained, in preparation for the support transfer, where larger angular momentum is required. Also, in the support transfer phase, faster speed is shown in fig. 4 (around the right loop), for the same reason as above.
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Although the number of data is a few, the plots looks diverge around the swing phase. This feature agrees that the degree of convergence of trajectory corresponds the volume of envelope, as is stated above.
Figure 2: superimposed plot of relationship between angles of left hip and left ankle
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Finally, we should mention that while this result is obtained from experiment of single subject and needs more data for further study, the data currently being processed seems to support the result obtained here.
variation exists even within the same motion
Finally, we describe a differentiation within same motion. We found that in some of trials (of without restriction condition) shows different pattern of trajectory, when momentum variable is considered. In fig. 5, a super imposed plot of the relationship between the head speed and the ankle is shown. This plot looks like fig. 3: although they are topologically same, the shapes are clearly different. In fact, what make divide these two cases is the number of trials. In the fig. 5, two last trial of ten trials are plotted and first eight trials are shown in fig. 3. This difference is not expected, since we did not recognise this small (but clear) difference at the experiment. Watching the animation of captured data, even in it is not easy to tell the difference. As an impression, in the latter case (i.e., fig5) the motion becomes smooth: we assume it is because of an “optimisation” process of a human through experience. Since higher speed is achieved around the support transfer in fig 5, the motion is smooth and regarded as efficient. Then we call the first eight trials as “unexperienced” rising and the last two trials as “experienced” rising. In this experiment it is shown that trajectory is variable even within an envelope. This feature is very important when one think about design of humanoid motion, since this variation may leads to the controllability within an envelope.
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Concluding Remarks
In the present work, through experimental study of a rising motion, we have revealed complex structure of the phase space of a rising motion and its relation to the global dynamics. Then we have shown that the degree of convergence of a bundle of trajectory corresponds the volume of envelope. Although this is expected, even within an envelope, variation is seen. This “microscopic” variation is unexpected. Since the envelope is regarded as a stable region, we expected convergence for all trials with small change as a result of experience. However, we have seen obvious differentiation. We think this differentiation may be due to localisation within an envelope. Since an envelope occupies a part of the phase space as a volume and its dimension is as high as that of the phase space. Then there are variation of trajectories for the same type of motion. Note that plots shown in this paper are two-dimensional and they are projection of original high dimensional manifold. Then study of a good method and choice of variables for projection is suggested as a future plan. Also, we need to a method to evaluate similarity between different pattern of coordinations. While
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we could distinguish several patterns, as general structure of nonlinear dynamics suggests [11], even smaller scale of microscopic structure may exist. We should note that by exploiting such microscopic structure, we may obtain controllability within an envelope to realise human-like flexible behavior.
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It is important to note that we do not focus on searching for an ideal trajectory based on some kind of optimisation. Human body is not a precision device and it is wise to chose a trajectory where stability is guaranteed by its own dynamics, such as within an envelope. A humanoid robot can be a precision device but the environment is not. Then it seems to be always a good idea to exploit a stable dynamical structure, whenever applicable. If we think about the relevance of our results for humanoid robotics, it is important to note that the dynamics discussed in the paper is that of a human’s body. The dynamics of human body includes muscle as elastic elements, which is not usually included in the robot’s mechanism. One possible difference is that volume of the envelope may be enlarged, as is shown in our former work [3]. To exclude this effect, we are going to feed the obtained trajectory to a dynamics simulator. However, we expect the essence, complex network of envelopes and nodes, is preserved. While so far we have discussed on the envelope, the node must be analysed in the future. Because experimental analysis of unstable structure requires much more data than that of stable structure and we could not investigate in this work. As mentioned above, numerical analysis by a dynamics simulator may be helpful. As for experiment, while we are processing the captured data, for the rising, this is much harder than other motion, because of frequent occulution of markers, quick rotation of body segments and high density of markers. This problem is expected to be solved by improvement of algorithm of motion capture system. In near future, we are going to establish the notion obtained here by collecting many data.
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Although detail mapping of the phase space is required, we think that by choosing “microscopic nodes”, fine tuning of behavior is possible. Then, the design of motion based on the global dynamics will be done in two stages: first choose rough trajectories by choosing nodes (i.e., macroscopic) and then tune the behavior by choosing microscopic nodes.
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ankle [deg] Figure 5: superimposed plot of relationship between head speed and angle of left ankle. No restriction is adopted for the foot placement. Two last trials of ten trials are shown.
To sum up, through the motion capture experiments of a human rising motion, we have shown complicated structure of the phase space. The degree of convergence within a bundle of trajectories corresponds the area of the envelope and this supports our
concept of the global dynamics. Also, investigation of the phase space suggests that microscopic structure exists, even in single envelope. By analysing and exploiting, it may lead to a flexible motion of a humanoid robot. acknowledgment The authors are greatly thankful for Drs. G. Taga, K. Kaneko for discussions. Also the authors are thankful for Mr Y. Kawamura and T. Nagasaki for assisting experiments. The concept of global dynamics originally suggested by Dr. A. Nagakubo and one of the authors (YK). This work is supported by COE program funded by STA (Science and Technology Agency of Japan).
References [1] K. Hirai, M. Hirose, Y. Haikawa, and T. Takenaka. The development of honda humanoid robot. IEEE International Conference on Robotics and Automation, 1998. [2] Yasuo Kuniyoshi and Akihiko Nagakubo. Humanoid as a research vehicle into flexible complex interaction. IEEE International Conference on Intelligent Robots and Systems, pages 811–819, 1997. [3] Tomoyuki Yamamoto and Yasuo Kuniyoshi. Harnessing the robot’s body dynamics: a grobal dynamics apporach. IEEE International Conference on Intelligent Robots and Systems, pages 518–525, 2001. [4] M. Vukobratovic and J. Stepanenko. On the stability of anthropomorphic systems. Mathematical Biosciences, 15:1–37, 1972. [5] Tad McGeer. Passive dynamic walking. Int. J. of Robotic Res., 9(2):62 – 82, 1990. [6] Tad McGeer. Passive walking with knees. Proc. IEEE Conf. on Robotics & Automation, pages 1640 – 1645, 1990. [7] J. Pratt and G. Pratt. Exploiting natural dynamics in the control of a planar bipedal walking robot. In Progeedings of the Thirty-Sixth Annual Allerton Conference on Communication, Control, and Computing, pages 739–748. 1998. [8] A. van Sant. Rising from a supuine position to erect stance. Physical Therapy, 68:185–193, 1988. [9] A. van Sant. Age differences in movement patterns used by children to rise from a supuine position to erect stance. Physical Therapy, 68:1330–1339, 1989. [10] Tomoyuki Yamamoto and Yasuo Kuniyoshi. Introduction of the etl/hh marker set for whole body motion analysis (in Japanese). proceedings of 19th annual conference of RSJ, pages 597–598, 2001. [11] V. I. Arnold and A. Avez. Ergodic Probrems of Classical Mechanics. W A Benjamin, 1968.
