alexa A Determinant Analysis to Detect the Singularity of the Extended Theo Jansen Mechanism in the Phase-Rotation-Amplitude Parameter Space | Open Access Journals
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Journal of Computer Science & Systems Biology
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A Determinant Analysis to Detect the Singularity of the Extended Theo Jansen Mechanism in the Phase-Rotation-Amplitude Parameter Space

Kazuma Komoda1* and Hiroaki Wagatsuma1,2
1Department of Brain Science and Engineering, Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu-Ku, Kitakyushu, Japan
2Laboratory for Dynamics of Emergent Intelligence, RIKEN Brain Science Institute, Hirosawa, Wako-shi, Saitama, Japan
Corresponding Author : Kazuma Komoda
Department of Brain Science and Engineering
Graduate School of Life Science and Systems Engineering
Kyushu Institute of Technology 2-4 Hibikino
Wakamatsu-Ku, Kitakyushu 808-0196, Japan
Tel: 81936956159
E-mail: [email protected]
Received November 19, 2015; Accepted December 14, 2015; Published December 21, 2015
Citation: Komoda K, Wagatsuma H (2015) A Determinant Analysis to Detect the Singularity of the Extended Theo Jansen Mechanism in the Phase-Rotation-Amplitude Parameter Space. J Comput Sci Syst Biol 9:1 010-022. doi:10.4172/jcsb.1000215
Copyright: © 2015 Komoda K, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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Abstract

Legged robots have a potential of being a walking machine on irregular ground. Eleven-bar linkages, Theo Jansen mechanism reproduces a smooth locomotion pattern as gait. Parallel motions have widely used in the heavy machinery and recently highlighted in a model of biological motions. The close-loop linkage simply provides a designed end-effector trajectory, whereas the trajectory is considered to be less modifiable due to the singularity problem. In the present study, the singularity on the modified Theo Jansen mechanism was addressed by introducing the parametric orbit as a new freedom point in the joint center, and analyzed its kinematics and dynamics by using multibody dynamics (MBD). The extendability of the mechanism in the viewpoint of flexibility in the gait trajectory was clearly demonstrated in the numerical simulation, providing new functional gait trajectories controlled by two control parameters that change the shape of the parametric oval in the joint center. In systematic determinant analyses of how broken trajectories were generated depending on four parameters, i.e., horizontal and vertical amplitudes and rotation angle of the joint center movement and its phase difference with the driving link, morpho-logical changes of generated trajectories in the phase-rotation-amplitude parameter space were revealed. Thus the extension capability of Theo Jansen mechanism was validated not only in smooth walking but also in jumping, climbing and running-like motions. In considering the ways of control, the present results indicated that there exists a inverse relationship between the rotation angle and the phase difference to significantly reduce the occurrence of the singularity and breakage failures of the mechanism, which is consistent with biological evidences of coupling oscillators that enables the nervous system to control the complex musculoskeletal system by using a few of simple parameters frequently represented by the phase and rotation in a torus state space.

Keywords
Closed-loop linkage; Theo Jansen mechanism; Multibody dynamics; Determinant analysis; Gait trajectory
Introduction
Flexible and adaptive movements generated from biological systems are of interest not only to academics in the area of biomechanics but also to engineers to improve traditional mechanical schemes. Closed loop linkages are widely known because of the parallel motions used in the Watt steam engine to transform the motion from parallel, Peaucellier- Lipkin and Hart’s inversor linkages to reproduce a completely straight line form cyclic motions by using different combinations of parallelograms. Bio-inspired mechatronics to mimic biological mechanisms in motions and behaviors have been investigated [1], and the legged motions are frequently modeled by the serial mechanism connected by joints with dampers and springs such as the passive walker [2], known as openloop linkages. The open-loop linkage is convenient for the generation of arbitrary trajectory of the end-effector with a supervision of every joint rotation angle governed by the central controller.
The bipedal locomotion is still a hot topic [3,4]; however the central controller paradigm either provides a heavy load of computation to calculate kinematics and dynamics of the system or requires a learning property of motion patterns. From the engineering viewpoint, it implies that the maximum payload relies on the torque performance of individual actuators embedded in joints of the open-loop linkage. An alternative solution is closed-loop linkages, which have been widely used such as mechanical transmissions of explosion power in engines to wheel crankshaft in car and the walking ability in uneven ground [5,6]. Interestingly, Theo Jansen proposed a kinematics for providing of legged robots in the form of the closed-loop linkage with eleven linkbars, called Theo Jansen mechanism and demonstrated that it worked for heavy payload conditions in a various type of similar mechanisms [7]. For analyses of the reproduced animal locomotion with smooth transition between swing and stand phases in a locomotion cycle, Giesbrecht [8] investigated an optimized orbit of legs from the viewpoint of a minimum energy input and a maxiimum stride length. Kim [9] demonstrated a real remote-controllable robot based on the mechanism to test the walking performance on shore and marsh place.
Theo Jansen aspired the ultimate gait trajectory and selected the best combination of linkage-bar lengths, which succeeded in providing a fixed locomotion pattern, whereas the mechanism stays in a position far from the alternative solution of legged systems based on the openloop linkage in the viewpoint of flexibility. In principle, closed-loop linkage has capabilities to change its motion even without changes of link-bar lengths due to additive motions of nodes. As efforts of extension of the original mechanism, Sitharam [10] investigated relationship between linkages by representing a connected component of the Euclidean realization space as a curve in a carefully chosen non-edge distance parameter space and Nansai [11] proposed a wide variety of gait cycle by adding actuator to part of linkages. These studies demonstrated extension capabilities of the Theo Jansen mechanism by heuristic methods, but it lacks the comprehension and structural method of how it is possible.
Deadlocks and the breakage of the system kinematics in an inappropriate design is inevitable in the closed-loop linkage, while multibody dynamics (MBD), which is a technique for describing mechanism mathematically, clears a path to investigate the issue in a vistaed wAy. In the present study, an additional movable node was selected from fixed joints of the original Theo Jansen mechanism according to the inspiration of the toe-knee-inguinal relation of the human leg, which was preliminarily reported in our previous publication [12] and then we hypothesized that functional gait trajectories involving climbing and jumping were extensively associated with motion patterns of the movable node assumed as the knee position. The issue to be addressed in this paper is to clarify of the limitation of extendibility of the Theo Jansen mechanism to provide new gait patterns and the elucidation of characteristics on constraints of the parametric space of the additional node movement given by the parametric oval orbit.
We introduced determinant analyses of the system kinematics and investigated a safety assurance to avoid breaking in the parametric space. Thus, the present paper is devoted to validate the hypothesis on the extendability of the Theo Jansen mechanism through the generation of new and unbroken trajectories of the closed-loop linkage by using the parametric oval obit in the additional node with following four parameters. The amplitudes of the moving range in horizontal and vertical Axes, the rotation angle of the oval orbit and the start position of the oval movement with respect to the synchronized movement of the input cycle rotation angle, i.e., the phase difference between the oval of the midpoint and the unit cycle of the input.
This paper is divided into the following sections: Section of “Hypothesis” addresses our hypothesis of how the original mechanism can be extended to multifunctional legged robots with a certain risk on the deadlock. Section of “Mathematical Formulations” describes the common framework of the model definition and preliminaries for multibody dynamics. Section of “Extension Model of the Theo Jansen Mechanism” describes the modeling formulation of a legged robot with closed-loop linkage known as the Theo Jansen mechanism. In Section of “Results of Numerical Simulations,” characteristic analyses of the extended Theo Jansen mechanism are discussed with temporal evolutions of the placement, posture, velocity, acceleration, and determinant in three different conditions. In Section of “Functional Trajectory Generation Focusing on Amplitude Change,” amplitude properties investigate for understanding behavior regarding adding orbits and leg’s trajectories. Phase differences distribution is described in one circle case and two ellipse cases by changing amplitudes and rotation angle in Section of “Functional Orbit Generation Focusing on the Phase Difference”. In Section of “Functional Orbit Generation Focusing on the Phase Difference”. distribution of nonbroken trajectories has been classified in the extended Theo Jansen mechanism for getting non-broken trajectories. Effective conditions are explained regarding amplitude, phase differences, and rotation angle in the extended motion in discussion to realize functional motion. Coordination of those conditions is the key for generating smooth gait trajectories like various animal motions based on kinematic constraints and the degree of freedom.
Hypothesis
The original Theo Jansen mechanism has two static joints: one is the circle center of the input torque generator O1 and the other is the meeting point of upper three components with two deforming rectangles and a fixed triangle O2 as shown in Figure 1 (a). Interestingly, we found a possibility of morphological changes of the gait trajectory if the second static joint O2 moves synchronously with the input circular movement around O1 with a specific range, direction and timing difference similar like the oscillation phase difference in biological locomotive mechanism, which was previously reported by Komoda and Wagatsuma [12,13] as illustrated in Figure 1 (b). Importance of phase differences among peripheral controller (nervous system) activities that govern motion timings of body components is known for its contribution to the generation of functional movements, such as walking with leg and handling with arms, called central pattern generator (CPG) [14]. Taga [15] demonstrated that a mutual coupling of CPGs successfully provides the robotic bipedal locomotion even in unpredictable environment by changing its locomotive patterns.
In addition, similar studies have been reported to extend from such a static walking patterns to dynamics motion even with jumping [16-18]. On the other hand, the phase difference was mostly discussed in the controller design and its protocols. It is difficult to find a significant notation of the phase difference in the mechanical design with kinematics and dynamics, except the Theo Jansen mechanism [7]. The present paper addressed a constraint problem to explore possible relationships in the space spanned by Axes of the amplitude, dominant direction and phase difference of O2 movement with respect to O1 motion, under the criterion of whether or not the closed-loop linkage provides unbroken motions. For this purpose, the parametric oval orbit was used for O2 movement conventionally. We hypothesized that the modified Theo Jansen mechanism has the capability to generate multiple types of the gait trajectories under certain condition represented by a subspace of the phase-rotation-amplitude parameter space, which exhibits a significant phase relationship. The amplitude limitation is trivial in the mechanical sense yet a phase-specific motion generation was a unique fact previously unknown, which suggests that the mechanical coordination in the closed-loop linkage is consistent with schemes of biological motions that reported in experimental evidences of CPG patterns [14]. Our proposed method of step-wise determinant analyses was employed to judge whether the motion is broken or unbroken to be a functional motion and the method was validated in the following sections.
Mathematical Formulations
Kinematics formulations with multibody dynamics
In order to analyze the certain moment reaction of a constrained dynamics system, it is necessary to describe the behavior of multibody system (MBS) by using the equation of motion. The MBS is constructed by a group of rigid and flexible bodies, which depends on kinematic constraints and some force parts. Kinematic constraints demonstrate linear or quadratic dependence on the generalized Cartesian coordinate. Various approaches for the generation of the equation of motion in the MBS can be found in various wAys [19-22].
If a planar mechanism is made up of nb rigid bodies, the number of planar Cartesian generalized coordinates is nc=3 nb. The vector of generalized coordinates for the systems is written as:
image (1)
Where image is the vector of planar Cartesian generalized coordinates for MBS.
A kinematic constraint between body i and body j imposes conditions on the relative motion between the pair of bodies at arbitrary joint k is obtained as:
image (2)
where ri is the vector to centroid of body, Ai is the rotation transformation matrix, and s' ki is the local representation of body fixed vector to point k.
According to the configuration of the MBS defined by n vector of generalized coordinates of q and t is the time, a set of kinematic constraint equations ɸ are obtained as:
image (3)
where ɸK (q) is the kinematic constraint equation, ɸD(q; t) is the driving constraints of MBS.
Equation (3) is related to time to obtain the velocity constraint equation. It introduces a second differentiation of Equation (3) with respect to time to obtain the acceleration constraint equation as:
image (4)
image (5)
where ɸq is the Jacobian matrix of the kinematic constraint equation, v is the velocity equation and γ is the acceleration equation.
Determinant analysis for detecting non-broken motions
Our purpose is to obtain functional motions on the extended Theo Jansen mechanism. It is important to clarify the position and posture for a specific purpose. Only if the mechanism generates a continuous orbit with the non-singular configuration throughout the orbit, we use it as the “functional orbit.” The consistency between the singularity and a part of angle in the mechanism becomes an effective method to detect an irregular orbit.
The singular configurations of a general n-linkage will be defined for which the legs position in a certain direction cannot be realized by any finite joint position [23], which means an impossible or broken motion. Let us consider n elements in the linkage and define the singular configurations using the Jacobian matrix as ɸq. Let n′ be given by :
image (6)
Then the singular configurations alwAys satisfy:
image (7)
In the Theo Jansen mechanism, n=n′ ≤ 36 and ɸq is square, a necessary condition for a q to be singular is given by:
image. (8)
In this analysis, the singular configuration is detected by the Jacobian determinant which is zero. If the determinant is equal to zero, the orbit breaks and it exceeds the normal joint range of motion. The extended Theo Jansen mechanism has two degrees of freedom (DOF) as redundant constraints, and it is necessary to find movable ranges before the lock-up.
In the original Theo Jansen mechanism, the gait trajectory consists of motion phases including stance and swing phases, which exhibit a continuous and smooth curvature shaped like an ellipse with a sharpen edge. The morphological change of the trajectory implies a generation of other functions beyond the walking as is illustrated in Figure 2, while discussions on the functionality needs to involve the ground condition. The inevitable problem before the functionality with ground conditions is how the mechanism can be extended without the lock-up, and this paper is devoted to mathematical analyses to evaluate the end effector trajectory with respect to the parametric space to control the additional O2 movement, which is called as the extended Theo Jansen mechanism. In order to obtain relationship an original motion and the extended one, we lead to mathematical description for getting limitation of the mechanism by using determinant.
A generated mechanism is classified as following two judgment conditions as shown in Figure 3. First, a determinant checks kinematical constraints in the mechanism. If determinant is equal to zero, a generalized mechanism becomes singular configuration. Second, a certain kinematical constraint is judged for checking bifurcations in the mechanism. In quadrilateral O2CED in Figure 4, if angle CO2D is more than rad or if angle O2DE is more than rad, then a generalized mechanism obtains a bifurcation in this linkage. Above judgment conditions represent a broken leg’s trajectory in our proposal mechanisms. Kinematics and dynamics of the original Theo Jansen mechanism and the possible extension form are clearly described through the MBD and the limitation of the mechanism is analyzed using determinant as described in the next section.
Extension Model of the Theo Jansen
Mechanism
According to the common framework of the model definition and preliminaries in Section III, a closed-loop linkage, the Theo Jansen mechanism can be treated consistently in multibody dynamics (MBD) methods, which means that a legged robot is constructed by individual differential algebraic equations to be able to analyze the placement, velocity, acceleration, and determinant.
Original Theo Jansen mechanism: The mathematical model of the Theo Jansen mechanism is described according to the kinematics formulation in the previous section. As is illustrated in Figure 4, vectors q with 36 elements including placements and attitude angles are represented in generalized coordinates:
image (9)
A set of kinematic constraint equations ɸ are given by Equation (3). The first thirty-five elements of the column matrix ɸK (q) are derived from kinematic constraint equa-tions. Whereas the last element ɸD(q, t) is derived by driving constraint equation, the equation of kinematic constraints and a driving constraint are obtained as:
image (10)
where l1 to l12 are link lengths, t is time, and ω is angular velocity of the driving link in the mechanism. Originally the Theo Jansen mechanism is known as eleven linkages, while in this case it is described twelve linkages in the MBD with generalized coordinates. Table 1 denotes a set of parameter values of the Theo Jansen mechanism. Therefore, the Jacobian matrix ɸq is obtained as:
image (11)
where it allows us to investigate placement, velocity and acceleration analyses kinematically.
Extension with parametric ovals: A few DOF are one good advantage obtained from the parallel linkage mechanism for generating a designed motion. According to our hypothesis, the mechanism can be modified by changing the static joint of O2 to the movable one, abbreviated as “joint center”, reminiscent of the knee point of the human leg. It is a partial extension of the DOF against the original mechanism toward additional motion generations, and the new motion to the joint center (O2) in Figure 4 is given as follows:
image, (12)
image (13)
Let Ax, Ay, ω, t, δx, δy, bx, and by be amplitude, angular velocity, time, phase difference, and Axis shift in x and y coordinates, respectively:
image (14)
image (15)
Equation (14) and Equation (15) describe an orbital shape in the leg. These equations, which have seven conditions for detecting one exact motion, are for general condition in the extended Theo Jansen mechanism. In the previous research, we found a functional leg’s orbit under a limited condition [12]. The phase difference between O2 and O1 and a vertical mobility of O2 were considered to be the clues for obtaining additional motions in the extended mechanism. In order to observe functional leg’s trajectory, the number of variables in Equation (14) and Equation (15) can be reduced by its dependency. By assuming that δx and δy are equal to the single variable δ, the equivalent phase difference is given by Equation (14) and Equation (15) in the condition that by is equal to Ay and bx is equal to zero. If by is less than Ay, the toe position exceeds the level of the ground, which means digging of the ground. Thus, we modified Equation (14) and Equation (15) as follows:
image, (16)
image (17)
The rotation matrix is used to change the rotation angle of the parametric orbit of O2 (the vertical longitudinal ellipse is assigned as 0 [rad]) is given as:
image (18)
where, ϕ is an arbitrary angle of the parametric oval. Furthermore, we added following two conditions for keeping a certain placement in O2. First, minimum O2 orbit point in y mechanism Coordinate placement is zero. Second, cross point between long Axis and short Axis in ellipse orbit O2 is zero in x coordinate. We expect that these conditions can assure the extended mechanism to prevent drilling in the ground by its legs. Thus, equation (18) is modified as following:
image (19)
Therefore, the relative coordinates drives are as follows, finally, the relative coordinates drive are given as follows:
image (20)
image (21)
image (22)
According to the above modification, the extended Theo Jansen mechanism has two degrees of freedom as redundant constraints. The mechanism can be investigated by changing four parameters Ax, Ay, ϕ, and δ to reveal what kinds motions are generated.
Results of Numerical Simulations
In this section, results of numerical simulations were presented as a consequence of the modeling discussed Section of “Extension Model of the Theo Jansen Mechanism.” As hypothesized in Section of “Hypothesis,” extended functional motions are able to be generated by adding a synchronous movement of O2 joint center, which represents a kind of knee position as an analogy so that the driving cycle of O1 and the end effector motion of G are assumed to be inguinal and toe, respectively [12]. For detailed analyses of such extended functional motions, the parametric oval movement of O2 was proposed as formulated in Section of “Extension with Parametric Ovals” and this section is devoted to the detailed examination of placement, posture, velocity, acceleration, and determinant of the extended Theo Jansen mechanism. Parameters and conditions for the analyses were described in Table 2, which is used in MATLAB based numerical simulation with the Newton-Raphson method. In following analyses, the constant angular velocity ω=0:5π rad/s (15 rpm) was assumed to be the driving link rotation of O1 consistently for every numerical simulation, as shown in Figure 4 consistently for every numerical simulations. In placement, curvature, velocity, acceleration and determinant analyses, the end effector, G which is the tip of the leg, is analyzed by using the generalized placement of the grounding link, and then the end effector placement of the Theo Jansen mechanism G=[Gx,Gy] was calculated by the center of twelve links [x12, y12]. In following subsections, prominent three motions were analyzed as functional movements for various locomotive actions especially for uneven fields.
Characteristic analyses on the extended Theo Jansen mechanism
Climbing-like motion: As an example of generated trajectory in the extended Theo Jansen mechanism using a set of parameters in the first column of Table 2, the temporal sequence of placements and postures during a cycle was obtained (Figure 5 (a)). The driving link input was given as the circular trajectory with the constant angular velocity ω (the circle at the origin of the figure) which is finally transformed into an elliptic trajectory as the leg’s trajectory including swing and stance phases of the gait. Since highest and lowest placements of the trajectory were representative points of swing and stance phases, the leg’s height was obtained by the subtraction of the minimum placement -0.61 m from the maxiimum placement -0.05 m to be 0.56 m from the ground. Thus, the generated trajectory was eight times higher than the height of the original gait trajectory and was 64.4% of the system’s total height (Figure 5 (b)). For the definition of the ‘climbing’ ability, gradeability should be considered as the maxiimum slope that a vehicle can climb in a stable condition, preventing slipping of wheels and legs on the ground and rollover that happens if the center of mass of the vehicle exceeds an acceptable range of the backward inclining, which is calculated by the line connecting of two peak points of the curvature (Figure 5 (c)).
The curvature is defined by using an edge of trajectory as image ,where x and y are placements of the leg’s trajectory. In the assumption that the friction of legs on the ground is high enough for stable onward movements, the most important issue is the acceptable range of the backward inclining, which is calculated by the relationship between the position of the center of mass and the endpoint of the contact area with the ground. In the case of legged robots, the contact area can be given by the gait trajectory. Ghassaei [24] investigated the position of the center of mass in the Theo Jansen mechanism through the integration of individual link analyses and concluded that it was located at just below the lower end of the driving link. According to the result, the maxiimum rotation angle is able to incline backward without rollover was about 45° and it can be considered as the acceptable range of the backward inclining in the Theo Jansen mechanism. The climbing-like motion of our result allows legs to lift up to 39.5° within the acceptable range, which is calculated by the slope of the line connecting inflection points forward and backward on the gait trajectory. According to the review of Seeni [25], wheel based robots can climb up to 25°, legged robots such as 8-legged system SCORPION [26] and 6-legged DLR Walker robot [27], can be estimate to be able to climb up to 35° and wheel-leg hybrid robots can climb higher than 35°. Hopping robots or hoppers are obviously suitable machine for climbing and hop-roll hybrids have wheels and legs in upper and lower surfaces to provide high capabilities beyond the concept of rollover accidents of vehicles. In comparison with those capabilities, our extended Theo Jansen has a climbing capability higher than wheel based robots and usual legged robots, and close to the wheel-leg hybrid robots [28,29].
According to the velocity analysis of the leg as shown in Figure 5 (d), minimum and maxiimum velocities were 0.04 m/s and 0.90 m/s, respectively and then the mean velocity was 0.42 m/s. The first peak appeared at the maxiimum speed of 0.90 m/s (t=1.278 s), the second peak appeared at the point with 0.52 m/s speed (t=3.222 s). As plotted on the trajectory, the first peak point was located at the midpoint in the swing phase and the second peak was located in the swing phase after passing through the highest placement of the trajectory. The trajectory shaped like a climbing-like motion for climbing obstacle and the leg moved quickly in the swing phase and slowly in the stance phase.
In the acceleration analysis in Figure 5 (e), there were two peaks at 1.55 m/s2 (t=0.844 s) and 1.49 m/s2 (t=1.600 s) in the plot of the temporal sequence of the absolute acceleration value of the leg movement. The two points were located respectively at the moment just after the lifting up the leg as an initiation of the swing phase and the moment just before the transition of the moving direction, i.e., the curvature peak, in the swing phase. This result indicate that the mechanism moves fast when the leg takes off from the ground accompanied with a complex acceleration control by mAximizing the acceleration at the midpoint of the swing phase and moves slow by minimizing the speed at the curvature peak point and then touches the ground by decreasing the acceleration to form the smooth trajectory.
In the determinant analysis as shown in Figure 5 (f), minimum and maxiimum determinants of the system were obtained as 1.16 × 10-6 and 4.60 × 10-6, respectively. When an angle was 107π/180 rad, a mechanism approached to a singularity posture. In this condition, a leg is moving swing phase for reaching edge in the trajectory. Local minimums of the determinant Da and Db during a cycle could not simply be correlated with characteristic points in the placement, curvature, velocity and acceleration.
Figure-of-eight motion: As a further example of the generated trajectory in the extended Theo Jansen mechanism, a figure-of-eight end effector trajectory was obtained using a set of parameters in the second column of Table 2, as shown in Figure 6 (a). Figure-of-eight trajectories were known to exhibit biological motions, such as swimming [30,31] and flying [32-34] even in microorganism [35]. Insect motions in flight realize smooth movements with a less energy consumption [32,33], which are easily generated by a combination of oscillatory components with amplitude and phase modulations known as Lissajous figures [36]. Therefore, mechanisms to generate such trajectories have been investigated in bio-inspired robotics [37-41]. On the other hand, figureof- eight trajectories were not recognized in the human locomotion before, while recent advance of three dimensional human motion analyses revealed the existence of figure-of-eight trajectories in gait if it is drawn to designate the root of the thigh as the origin [42].
In fact, the human locomotion generated by a complex human musculoskeletal system is not simply equivalent to the insect motion generation. Interestingly, as Ivanenko [43] demonstrated, there exists a consistency of modular based control mechanism to reduce the freedom of degrees using a synchronous coordination among modular units, which generates a movement-speed dependent change in the gait trajectory. They classified gait trajectories into walking, running, hopping, crouched walking, obstacle avoidance and air-stepping, and found a figure-of-eight trajectory in the case of running.
In the present study, the height of the figure-of-eight trajectory was 0.25 m from the ground (29.6% against the total height), which suggested a non-trivial relationship with the human locomotion.
According to the velocity analysis of the leg as shown in Figure 6 (d), minimum and maxiimum velocities were 0.04 m/s and 0.57 m/s, respectively and then the mean velocity was obtained as 0.30 m/s. The first peak appeared at the maxiimum speed of 0.57 m/s (t=1.556 s), the second peak appeared at the point with 0.42 m/s (t=3.633 s). As shown by the plot of trajectory, the first point is located at the midpoint in the swing phase and the second peak was located at the midpoint in the swing phase after passing through the highest placement of the trajectory. The trajectory shaped like a figure-of-eight motion for running slowly. In addition, the leg moved quickly in the swinging up motion and slowly in the swinging back motion.
In the acceleration analysis in Figure 6 (e), there were two peaks of points 0.85 m/s2 (t=1.000 s) and 0.82 m/s2 (t=2.067 s), having a similar value. In the determinant analysis as shown in Figure 6 (f), minimum and maxiimum determinants of the system were obtained as 3.50 × 10-6 and 4.60 × 10-6, respectively. The minimum determinant was obtained when the angle was 128π/180 rad, while it was far from a singularity posture as values larger than 3.4 × 10-6.
Dropping-down motion: As the final example, a dropping–down motion was generated via the extended Theo Jansen mechanism using a set of parameters in the last column of Table 2, and placements and postures during a cycle were illustrated in Figure 7 (a). It is difficult to find a simple coincidence between the dropping–down motion with biological motions to realize a specific function, while it should be focused on the compensation property. The generated motion exhibiting as a small circular orbit was obtained by the cancellation of two types of motions, so that the joint center O2 moves to offset the driving link cyclic motion O1 inside the mechanism by anti-phase cyclic motion between two cycles. Indeed, the trajectory height was reduced as 0.13 m from the ground (15.6% of the total height).
According to the velocity analysis (Figure 7 (d)), minimum and maxiimum velocities were 0.004 m/s and 0.13 m/s (the mean velocity was 0.08 m/s), which were quite small in comparison with above two conditions. The first peak was 0.90 m/s (t=1.278 s), and the second peak was 0.52 m/s (t=3.222 s). In the acceleration analysis (Figure 7 (e)), a similar tendency appeared as small values with three peaks as 0.18 m/s2 (t=0.522 s), 0.22 m/s2 (t=2.233 s) and 0.20 m/s2 (t=2.756 s).
In the determinant analysis (Figure 7 (f)), minimum and maxiimum determinants of the system were obtained as 2.42 × 10-6 and 4.39 × 10-6 respectively. When the angle was 235π /180 rad, the minimum determinant was obtained, which was not critical and apart from a singularity posture.
In Figure 8, characteristic properties of the acceleration during a cycle were analyzed in comparison of the three generated trajectories obtained from the extended Theo Jansen mechanism and an original trajectory. The line width represented the amount of the acceleration, which is proportional to the force generation. By using the parametric orbit defined in O2, the original gait trajectory (Figure 8 (a)) was successfully modified to a climbing-like motion (Figure 8 (b)), a figureof- eight motion (Figure 8 (c)) and a dropping-down or less mobility motion (Figure 8 (d)). As represented as the line width, a consistent acceleration change was found in Figure 8 (a)-8 (c) as the twisted shape, which included a sharp negative peak of the acceleration change. Such an acceleration property acted as the quick take-off from the stance phase to the swing phase and the soft grounding from the swing phase to the stance phase, which realized a smooth gait trajectory. In all three cases, the minimum determinant marked as Da coincidentally appeared just before the negative peak. The result showed that the consistency through the trajectory modification associated with the O2 movement preserved the properties for generating a smooth gait trajectory even in modified ones.
Functional trajectory generation focusing on amplitude change
In the previous section, contribution of the additional movement in the joint center O2=[xO2, yO2] for modifying the trajectory of the end effector was demonstrated and some functional motions were found based on the systematic parametric change. On the other hand, analysis of the robustness to pre-vent the generation of broken motion or dead lock is necessary in accordance with the criterion defined in Section of “Determinant Analysis for Detecting Non-Broken Motions.” In this section, characteristic analysis of the generated trajectory was investigated by focusing on amplitudes of the parametric orbit O2. According to Equation (16) and Equation (17), a set of four parameters that consist of horizontal and vertical amplitudes Ax and Ay, the rotation angle applied to the O2 orbit as ϕ, and the phase difference determining what kind of the new trajectory is generated. In this section, generated trajectories were analyzed by focusing on the amplitudes Ax and Ay from 0 to 350 × 10-3 m using parameters of Table 3.
In the case of the Ax variation, various trajectories were examined with the horizontal amplitude Ax changing from 0 to 350 × 10-3 m associated with the change of the rotation angle ϕ under the fixed condition of the vertical amplitude Ay=100 × 10-3 m. Figure 9 (a) illustrated 96 types of the O2 parametric orbit by the combination of eight amplitudes of Ax and twelve rotation angles, which were plotted in concentric circles with respect to those variations, and Figure 9 (b) showed resultant trajectories of the end effector in the same arrangement. Black dots showed a starting point in the joint center O2 moving in the anti-clockwise direction. Generated trajectories included oval, halfmoon, and figure-of-eight non-broken shapes. Broken trajectories were plotted as grAy lines and serious broken trajectories were omitted from plot (the movement exceeded over 1 m2 area). The serious broken types were denoted by cross and star marks after the classification using the criterion defined in Section of “Determinant Analysis for Detecting Non-Broken Motions” and illustrated in Figure 3 (the cross and star marks represent respectively the determinant condition failure and angle condition failure). Fine trajectories of 34 types were obtained and the success rate was 35.42%. According to the change of the horizontal amplitude Ax, generated trajectories varied from small shapes to large shapes. Non-broken trajectories Ax obtained was from 0 to 250 × 10-3 m, when ϕ was equal to π/6 and 2π/6 rad. According to the change of ϕ, generated trajectories were demonstrated in a certain angle range as -2π /6 ≤ ϕ ≤ 2π/6 rad. On the other hands, broken trajectories were generated when angle ranges as π/2 ≤ ϕ ≤ 7π/6 rad.
In the case of the ϕ–δ variation under the fixed condition of the vertical amplitude [Ax, Ay]=[100, 50] × 10-3 m, parameters in the last column of Table 4 were used as ellipse stretched in the normal direction against the concentric circle. Figure 12 (c) illustrated 144 types of the O2 parametric orbit by the combination of twelve steps of and twelve δ rotation angles ϕ, which were plotted in concentric circles with respect to those variations, and Figure 12 (d) showed resultant trajectories of the end effector in the same arrangement. Similar types of generated trajectories were obtained as well as the previous analyses. Fine trajectories of 71 types were obtained and the success rate was 54.17%. According to changing ϕ, generated trajectories were obtained as regular patterns. According to the change of δ and ϕ, generated trajectories varied as cyclic patterns. Non-broken trajectories were obtained, when δ-ϕ is from π/6 to 7π/6 rad.
In Figure 12 (d), trajectories with ϕ=π/2 rad were examined as a sample to analyze the sensitivity with δ (bordered by the dotted line) and the detail analysis was shown in Figure 12. The detail analysis of the selected parameter range was shown in Figure 13. In this case, broken trajectories were found in the range [π/6, 4/6π] rad while non-broken trajectories were obtained in others, which represented the δ sensitivity. In the comparison with the amplitude modification, which had a scattered distribution of non-broken trajectories and thus is unable to specify a consistent area, this result indicates that the modification using the parameter δ can be used for systematic change of the non- broken trajectories, avoiding singular configurations.
Systematic analyses in four controllable parameters
As the final analyses, we focus on the discussion of the success rate to generate non-broken trajectories from the extended Theo Jansen mechanism in the viewpoint of four parameters with amplitudes Ax and Ay, rotation angle of the O2 movement ϕ, and the phase difference δ between O2 and O1 movements. In this section, non-broken trajectories were analyzed by changing Ax, Ay, δ and ϕ respectively according to Table 5.
With respect to parameters of amplitudes Ax and Ay varying from 0 to 350 × 10-3 m, total 144 types conditions were examined whether they were broken or not to calculate the success rate of the condition. Figure 14 showed the success rate of every condition. White pixels represent the complete success condition without any broken trajectories and black pixel represents complete failure with broken trajectories only. Edges defined by 75% success rate was found around amplitudes Ax and Ay equal to 40 × 10-3 m, 58% success rate was found around Ax=Ay=100 × 10-3 m, 42% success rate was found around Ax=Ay=110 × 10-3 m, 25% success rate was found around Ax=Ay=230 × 10-3 m, and 0% success rate was found around amplitudes Ax and Ay equal to 260 × 10-3 m.
With respect to parameters of the rotation angle ϕ and the phase difference varying from 0 to 11π/6 rad every π/6 rad, total 906 types’ conditions were examined whether or not broken to calculate the success rate of the condition. Figure 15 showed the success rate of every condition. White pixels represent the complete success condition without any broken trajectories and black pixel represents complete failure with broken trajectories only. The whitest pixel represents the success rate with 74%.
In this systematic analyses of how the extended Theo Jansen mechanism generates non-broken trajectories avoiding the singularity configuration, the parametric oval of the joint center O2 was validated and, as the new finding, necessary conditions not only rely on the limitation of the amplitudes of the oval size but more significantly rely on the linear relationship between the rotation angle ϕ and phase difference as shown in Figure 15 exhibiting an inverse proportion.
Discussion
Similarity and dissimilarity with the biological motion
In the present analysis, the extensibility of the Theo Jansen mechanism was analyzed using multibody dynamics [19,20] and the reproduction of motions that extensionally-defined from the original walking pattern was demonstrated by releasing the lock of the joint center O2 and introducing the parametric oval.
As is discussed in Section of “Results of Numerical Simulations,” insect motions have been widely analyzed [32,33], and applied to robotics known as bio-inspired robotics [38-40,42]. In the analysis of the human motion trajectory, Ivanenko [43] investigated various gait trajectories and decomposed them into principal components, which enabled to reproduce the original trajectories, such as walking, running, hopping, crouched walking, obstacle avoidance, air-stepping and figureof- eight motions in running. Theo Jansen mechanism was highlighted in the similarity of the biological motion, which is still unclear what kinds of characteristics are similar [24]. In the viewpoint of similarity to the trajectory shape, the extended Theo Jansen mechanism can provide a figure-of-eight trajectory in gait, while the linkage mechanism is designed by linkages based on the mechanical engineering which is not simply corresponds to the human musculoskeletal system. It is quite different in the viewpoint of the physical property. Interestingly, the present result revealed that a specific relationship between the O2 and O1 as the phase difference δ and rotation angle ϕ provided a simple set of variables to enable the system to be controlled for generating various functional motions even if it was the complex system. Assumptions of the biological mechanism to control body parts in the phase-rotationamplitude parameter space was frequently discussed in the model using phase-coupling oscillators having synchronization and body coordination [15]. Such a control mechanism has the capability to generate an extended behavior by adding a new component of the synchronous oscillator [18]. The present result demonstrates the importance of the coupling property coordinating between the body and the controller (nervous system) through the phase and rotation even in a physically different linkage system.
Singularity analysis in the linkage system
Singularity analyses have been investigated in the parallel linkage system especially for planer linkages and manipulators [44-50]. A geometrically and physically based method for the detection of singularities was proposed [48], which provided information representing the distance of a given position to singularities. In the planer robot with multi legs, the singularity detection associated with a robot controller is useful for a path planning algorithm avoiding singularities during the locomotion. In order to find an appropriate path, the workspace can be defined as a virtual potential field [48,49] and such a method can be applied to the real-time control.
In a planar parallel manipulator, Gosselin and Angeles [44] established the mathematical foundations with Jacobian matrices of the chain and demonstrated the applicability to planar RRRP mechanism, the Watt linkage, planar three-degree-of-freedom manipulator with revolute actuators and spherical three-degree-of-freedom parallel manipulator. The extended Theo Jansen mechanism can be considered as an extension of the Watt linkage and our method of the singularity detection has a further theoretical perspective with respect to their method. Focusing on the geometrical property, Liu [47] proposed a method for detection of the singularity in parallel manipulators, by using differential topology strongly influenced by the Morse theory [51]. Morse theory analyzes the topology of a manifold by focusing on differentiable functions on the target manifold, and it allows the target to find a specific topological structure and handle decompositions on manifolds based on their homology, like a part of the torus as a simple example. In the present study, we found the phase-rotation relationship in the parameter space to detect the singularity, which can be considerable in the topological feature extracting from descriptions using generalized coordinates based on the physical context of the kinematics and dynamics.
Conclusion
In the present study, multibody dynamics was introduced to investigate the singularity of the extended Theo Jansen mechanism with respect to the parametric oval given as a new freedom point at the joint center O2. The method realized the system to exhibit the extendibility in terms of flexibility in the gait trajectory, including climbing-like, figureof- eight and dropping-down motions. In the systematic numerical simulation, the extended Theo Jansen mechanism reproduced various new functional gait trajectories, associated with four parameters with amplitudes of the parametric oval, its rotation angle and the phase difference between the joint center O2 and the driving link O1 movements, coupling with the proposed determinant analysis to detect the singularity and dead lock of the system. Importantly, a specific inverse proportional relationship between the rotation angle ϕ and phase difference δ in the contour map of δ–ϕ dependencies of nonbroken trajectories is established, which csts of individual success rates under every condition. To prevent the singularity of the extended Theo Jansen mechanism, necessary conditions not only rely on the limitation of the amplitudes of the oval size but more rely significantly on the phase–rotation relationship.
Acknowledgements
This work was partly supported by Council for Science, Technology and Innovation, ‘’Cross-ministerial Strategic Innovation Promotion Program (SIP), Infrastructure Maintenance, Renovation, and Management’’. (funding agency: NEDO)
References

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