TRBR: Flight body posture compensation for transverse ricochetal brachiation robot

Mechatronics 65 (2020) 102307

Contents lists available at ScienceDirect

Mechatronics journal homepage: www.elsevier.com/locate/mechatronics

TRBR: Flight body posture compensation for transverse ricochetal brachiation robot Chi-Ying Lin a,b,∗, Zong-Han Yang a a b

Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 10672, Taiwan Center for Cyber-physical System Innovation, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 10672, Taiwan

a r t i c l e

i n f o

Keywords: Transverse ricochetal brachaition robot Swing phase Flight phase Body posture compensation Arm swing motion

a b s t r a c t Transverse ricochetal brachiation is a sophisticated locomotion style that mimics athletes swinging their bodies with their hands on a ledge in order to propel themselves for a leap to a target ledge. This paper describes the development of a transverse ricochetal brachiation robot (TRBR) and outlines motion control strategies for active flight body posture compensation. The crucial design parameters were obtained by formulating an optimization problem with the goal of maximizing flight distance. Shoulder joints with switchable stiffness were used to enable resonance excitation via the swinging of a robot tail during the swing phase, while enabling tight arm-andbody engagement during the flight phase. Novel electric grippers were designed to provide the required holding forces as well as quick-release functionality to ensure that the kinetic energy accumulated during the swing phase could be transferred smoothly to the flight phase. The reference trajectory of the robot tail was obtained using an optimization procedure based on a dynamic model of the swing phase. We also adopted a dynamic model for the flight phase to elucidate the effects of midair body rotation with the aim of developing body posture compensation methods. Simulation and experimental results demonstrate the efficacy of the proposed body posture compensation method based on a successive loop closure design in improving flight body posture during transverse ricochetal brachiation. The integration of arm swing motion with tail compensation also proved highly effective in enhancing hang time and travel distance.

1. Introduction Brachiation robots mimic the movements of primates that use their limbs to swing from branch to branch [1–3]. The locomotion of these robots can be categorized as continuous brachiation [4,5] or ricochetal brachiation [6]. Continuous brachiation is generally used in cases where the gaped distance is within hand-holding range, whereas ricochetal brachiation is generally used in cases where the gaped distance exceeds hand-holding range. Most robots that use continuous brachiation employ a two-link mechanism [7] and swing control strategies aimed at providing precise tracking control of end-effectors (grippers) [4,8–13], efficient energy consumption [7,14–16,29], and transition schemes to reduce the impact of forward grabbing [15]. A variety of motion planning and control strategies have been proposed to investigate brachiation performance under various hand-holding conditions, including regular branches [15–19], irregular branches [20], moving branches [16], and flexible bars [21,22]. Research into ricochetal brachiation robots (RBRs) has focused mainly on the development of locomotion models based on high-DOF

dynamic systems [6,24] and simulations to facilitate motion planning and control for two-link robots [23,25,26]. Little research has gone into the mechatronic design and control methods for ricochetal brachiation. Results obtained from previous experimental studies [27,28] give some indication of the difficulties this entails. (1) It is difficult to predict timing for the grabbing of target hand holds due to the effects of modeling errors and air resistance during this short flight time. A gripper must be designed to compensate for uncertainties in the grabbing motion. (2) All existing motion control methods to adjust the midair posture of the robot were developed using a simplified 2D model [23,25,26,31,32], despite the fact that an unbalanced body mass and external disturbances can introduce undesired body rotations during the actual 3D flight phase [30]. (3) Providing the desired locomotion as well as the ability to switch from swing phase to flight phase for leaping makes the mechatronic design and implementation a non-trivial task. The issues mentioned above were encountered in the development of transverse ricochetal brachiation robots (TRBRs) [27,28]. The term

∗ Corresponding author at: Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 10672, Taiwan. E-mail address: [email protected] (C.-Y. Lin).

https://doi.org/10.1016/j.mechatronics.2019.102307 Received 2 February 2019; Received in revised form 3 September 2019; Accepted 23 November 2019 0957-4158/© 2019 Elsevier Ltd. All rights reserved.

C.-Y. Lin and Z.-H. Yang

“transverse” highlights the difference between the movement of this type of ricocehtal brachiation robot and conventional RBRs. The design of TRBRs is based on the locomotive style of athletes swinging their bodies transversely with their hands on a ledge in order to propel themselves into a leap to a target ledge [27]. The principles of TRBR locomotion lie in a smooth transition from the swing phase into the flight phase, and then into the landing phase. TRBRs can be applied to climbing tasks requiring movement between ledges on a wall, such as exterior window sills or eaves. Existing RBRs were designed to climb horizontal ladders with large gaps between the bars [23]. First-generation TRBRs (TRBR-I) [27] comprised two arms, a body, and a tail. A parallelogram mechanism formed by the arms and body can help to maintain body posture during the swing phase and reduce the accumulation of excessive angular momentum, which could cause body rotation during the flight phase. We derived a simplified 2-DOF dynamic model for the swing phase to empirically determine the resonant frequency with which to formulate a swing strategy for ricochetal brachiation. Our preliminary results on swing excitation and one-hand grabbing motion demonstrated the feasibility of the prototype based on a pair of pneumatic grippers. Robots that rely on the simultaneous actuation of arm and tail motors tend to be inefficient in accumulating kinetic energy, due to the effects of friction associated with geared mechanisms at the shoulder joints. This issue was addressed in the design of TRBRII [28] by integrating electromagnetic clutches between arm linkages and actuating motors. We implemented a switchable shoulder stiffness scheme to eliminate all stiffness during the swing phase and thereby promote the efficient accumulation of kinetic energy (using only tail swing) and then impose high stiffness during the flight phase to facilitate arm posture control. We integrated an electrically-driven robotic gripper with a trigger mechanism as an alternative to pneumatic grippers, thereby allowing tight handhold capability during the swing phase and rapid hand release capability at the moment of transition from the swing phase to the flight phase. We developed a model-based optimization method to derive the parameters that would ensure the desired tail reference trajectory in order to improve swing excitation efficiency. We also developed a dynamic model of the flight phase by which to analyze the midair body posture and assess the feasibility of using full tail rotation for posture compensation. Unfortunately, the simple open-loop compensation method developed in this study is susceptible to inefficient control energy consumption and poor robustness against unknown disturbances. In this paper, we describe the development of a novel TRBR with improved mechanical design (TRBR-III) and a variety of strategies to actively control body posture during the flight phase of transverse ricochetal brachiation. Robot design parameters were obtained by formulating model-based optimization problems under specific system constraints to minimize the size of the robot and expand the leaping distance. We employed a controlled arm swing strategy during the flight phase to extend the transverse distance travelled by the grippers and improve the likelihood of holding the target ledge, without having to increase the leap distance of the center of mass (COM) of the robot. Experiments on ricochetal brachiation were conducted to evaluate the efficacy of the proposed robot design and analyze the performance of the proposed posture compensation methods. The contributions of this paper are summarized as follows: (1) The design parameters of the robot and gripper were optimized to enhance the efficiency of resonant excitation to provide a more reliable hand-release between complex gait transitions. (2) We developed a motion control scheme based on successive loop closure (SLC) [33] to deal with the issue of body posture compensation in an underactuated robot during the flight phase. (3) We developed a strategy incorporating arm swing with tail swing to extend leaping distance, as a demonstration of the potential of using arm-body-tail brachiation robots for novel forms of locomotion styles.

Mechatronics 65 (2020) 102307

The remainder of this paper is organized as follows. Section 2 presents our problem statement explicitly identifying the design goal of the proposed robot. Section 3 identifies crucial design parameters and details pertaining to the mechatronic implementation of the proposed robot. Section 4 reviews a dynamic model of the robot during the flight phase. Section 5 introduces the methods proposed for body posture compensation during flight and corresponding simulation results. Section 6 presents experiments involving ricochetal brachiation. Concluding remarks are presented at the end of the paper.

2. Problem statement As shown in Fig. 1, the process of transverse ricochetal brachiation can be broken down into a swing phase, a flight phase, and a landing phase. Accumulating the energy required for leaping requires swinging the tail repeatedly at the resonant frequency of the robot. After releasing the hands, the robot must maintain appropriate posture in preparation for a two-handed landing. The above process is deemed successful only if the robot comes to a stable stop on the target ledge without falling to the ground. In this study, we obtained the tail trajectory for resonant excitation using the model-based optimization method reported in our previous work [28]. Accumulating energy for jumping requires that the grippers provide clamping forces sufficient to minimize shaking effects that would otherwise dissipate swing energy. The speed of release by the grippers can also affect the success of transition from swing phase to flight phase. Releasing the grippers rapidly can help to minimize the loss of kinetic energy and makes it easier to coordinate the timing for the leap. Stiffness in the shoulder joints should be minimized to prevent resistance that might otherwise lead to energy loss during the swing phase. As shown in Fig. 1, angular momentum accumulated during the swing phase can lead to flipping along the yaw axis, which could negatively affect the posture after transition to the flight phase. Roll rotation caused by a difference in the speed of release between the grippers or unbalanced mass in a multiple linkage configuration could also bring the grippers out of the vertical swing plane, i.e., out of alignment with the final hand-holding action. Our primary focus in this study was to eliminate or overcome the effects of yaw rotation and roll rotation. Yaw rotation during the flight phase was minimized by actively adjusting the body posture of the robot. Roll rotation was minimized by passively adjusting the center of mass of the mechanical and electronic components of the robot. We employed switchable stiffness to activate the electromagnetic clutches. This ensures that the body and arm links remain connected rigidly together at the shoulder joints, i.e., without free rotation [28]. In so doing, we were able to simplify the problem of control by examining two robot postures in the brief period before landing: (a) a tilted body posture; and (b) a vertical body posture. We assumed that the center of mass of the robot was at the same height in both cases. We also assumed that the target ledge was the same size as the starting ledge and at the same elevation from the ground. In Fig. 1, 𝜃 b refers to the angle between the centerline of the robot body and the horizontal. We can see in that in the case of posture (a), the left arm of the robot will bump into the target ledge first, resulting in an earlier (off-balance) landing. In the case of posture (b), a longer hang time will result in a longer overall leap distance, despite the fact that the distance traveled by the COM of the robot is the same. Our objective in this study was to maintain the vertical body posture (𝜃 b = 90°) during the flight phase of the maneuver to facilitate a stable landing. In the following, we list issues that must be considered in the design of the mechatronic system for transverse ricochetal brachiation: (1) The grippers must provide high clamping force and quick release capability.

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 1. Process of transverse ricochetal brachiation: (a) tilted body posture; (b) vertical body posture.

We first applied the equation of projectile motion to derive the flight trajectory of the robot (COM) in order to formulate an optimization problem pertaining to leap angle 𝜃 j . Problem formulation was based on the assumption that the work performed by the motor between the moment of reaching the maximum swing amplitude and the moment of leaping was equal to the amount of energy dissipated during that period. Based on the conservation of energy, the kinetic energy available for leaping is equal to the difference between the potential energy at the moment of reaching the maximum swing amplitude and the potential energy at the moment of leaping. Leap velocity vj can be obtained as follows: √ ( ) 𝑣𝑗 = 2𝑔 𝑙𝑎 cos 𝜃𝑗 − cos 𝜃max (1) where g refers to gravitational acceleration. Assume that the robot leaps between two parallel ledges at the same distance from the ground. Based on the fact that the flight time before the robot grabs the target ledge is 𝑡 = 2𝑣𝑗𝑦 ∕𝑔, we can use the equation of projectile motion (2) to derive the transverse flight distance xj as a function of la and 𝜃 j , as follows: Fig. 2. Simplified diagram showing ideal realization of transverse ricochetal brachiation.

(2) The swing amplitude should be large enough to store kinetic energy sufficient for leaping. (3) Roll rotation should be minimized to prevent the robot from flipping beyond the vertical swing plane. (4) A vertical body posture should be maintained until the end of the flight phase to prevent an off-balanced landing. 3. Development of transverse ricochetal brachiation robot In this study, we adopted the arm-body-tail robot design proposed in the previous generations of TRBRs [27,28]; however, we made a number of fundamental changes to improve locomotion performance. In this section we analyze the design parameters, the configuration of the mechatronic systems, the design of the gripper within the context of the design requirements, and sensory and control configurations. 3.1. Analysis of design parameters Once sufficient swing energy has accumulated, the robot is designed to release its grippers to initiate transverse ricochetal brachiation. Fig. 2 presents a simple illustration showing the ideal realization of this locomotion scheme and Table 1 lists the parameters. Assume that the robot releases the grippers immediately upon reaching the maximum swing angle 𝜃 max . To simplify our analysis, let us assume that after releasing the grippers, the two arms swing in a clockwise direction until they are perpendicular to the target ledge. This hand-up posture is then maintained until landing is complete.

1 2 𝑔𝑡 2 𝑣jx = 𝑣𝑗 cos 𝜃𝑗 , 𝑣jy = 𝑣𝑗 sin 𝜃𝑗

(2)

( ) 𝑥𝑗 = 4𝑙𝑎 cos 𝜃𝑗 − cos 𝜃max cos 𝜃𝑗 sin 𝜃𝑗 .

(3)

𝑥 = 𝑣jx 𝑡, 𝑦 = 𝑣jy 𝑡 −

The desired flight distance satisfying the requirements mentioned in Section 2 is denoted as xjd , which is the sum of the transverse distance traveled by the robot arm to reach hand-up posture (xs ) and the flight distance of the robot COM (xj ). 𝑥jd = 𝑥𝑠 + 𝑥𝑗 , 𝑥𝑠 = 𝑙𝑎 sin 𝜃𝑗 .

(4)

The above equations indicate that increasing the length of the robot arm would increase the overall flight distance; however, this would also require greater torque for the arm linkage to engage with the arm motor when activating the electromagnetic clutches. Thus, we must consider the constraints on the torque provided by the clutch 𝜏 a and the maximum swing angle 𝜃 max when deriving parameters la and 𝜃 j in (4). Considering that the maximum angular velocity of the robot arm occurs at the moment of leaping, we can derive a simplified constraint condition for 𝜏 a . Let 𝜃 max = 90° and 𝜃 j = 0°. Thus, using Eq. (1), the leap velocity vj can be written as follows: √ 𝑣𝑗 = 2𝑔 𝑙𝑎 . (5) Previous researchers have suggested a robot leap time between 0.1 s and 0.2 s [27,28]. Assume that the robot arm halts its rotation within 0.1 s after leaping. Then the angular velocity 𝜔j and angular acceleration 𝛼 j of the robot arm can be expressed as follows: √ √ 2𝑔 𝑙𝑎 10 2𝑔 𝑙𝑎 𝜔𝑗 = , 𝛼𝑗 = . (6) 𝑙𝑎 𝑙𝑎

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Table 1 Parameters used in Fig. 2. Symbol

Meaning

mr ma mb mt 𝜃 max 𝜃j ls la lt lg hj

mass of entire robot (center of mass) lumped mass of one arm mass of the body (center of mass) lumped mass of the tail maximum arm swing angle leap angle distance between two arms length of the robot arm length of the robot tail gap between two ledges height difference between the time at which the maximum swing amplitude is reached and the time at which the grippers are released to initiate the leap leap velocity of mr x component of vj y component of vj transverse distance traveled by robot arm before reaching hand-up posture after releasing grippers transverse flight distance of mr

vj vjx vjy xs xj

We employed a point mass model to estimate the holding torque around the shoulder joint, i.e., the location in which the clutches are installed. Under this model, the moment of arm inertia is estimated as 𝐼𝑎 = 𝑚𝑎 𝑙𝑎2 and the torque required to prevent the shoulder joint from rotating, 𝜏 a , can be calculated as follows: √ 𝜏𝑎 = 𝐼𝑎 𝛼𝑗 = 10𝑚𝑎 𝑙𝑎 2𝑔𝑙𝑎 , 𝜏𝑎 ≤ 𝜏𝑐 (7) where 𝜏 c = 0.06 kg·m is the torque specified for the clutches (Chain Tail Co. Ltd., MCFS06AA). Based on previous design experience [27,28], we set the mass of the arm at 0.05 kg and the length at 0.09 m. Integration based on the constraint where 0 ≤ 𝜃 j ≤ 𝜃 max , we obtained the following optimization problem: { ( ) } 2 arg max 𝑥𝑗𝑑 𝜃𝑗 subject to 0 ≤ 𝜃𝑗 ≤ 𝜃max (8) 𝜃𝑗

where parameter 𝜃 max in (8) is set at 80° to accommodate the structural constraints of the mechanisms. The results obtained from (8) indicate that the maximum leap distance is 16.29 cm when using an arm length of 0.09 m and leap angle of 39.34°. We then sought to determine tail length lt and tail mass mt using the simulation results obtained for the swing phase. We first estimated the tail swing frequency for resonant excitation (denoted as fe ) using a simplified one-link pendulum model, which can be expressed as √ 𝑔 1 𝑓𝑒 = = 1.66 Hz (9) 2𝜋 𝑙eq where leq is the equivalent length of the one-link pendulum. With the tail swing frequency fixed at 1.66 Hz, the swing angles of the arm can be analyzed by altering the values of lt and mt , in which a proportionalderivative (PD) control law is used to track the reference trajectory of the tail based on the dynamic model derived in [27], as shown in Fig. 3. The moment of inertia of the tail link is determined by the length and mass of the tail; therefore, a larger moment of inertia provides more swing energy and a larger swing angle for leaping. We sought to minimize the size of the robot in order to preserve actuator efficiency associated with the weight of the tail by treating the tail as the lower portion of the human body when determining the mass of the robot tail based on the anthropometric database reported in [34]. In the example in Fig. 3, an arm swing angle of 𝜃 as = 90° can be achieved by using a robot body mass mb of 1 kg, a tail mass of 220 g, and a tail length of 0.09 m. A robot design with smaller mb requires less swing energy; therefore, the mass of the body should be kept as small as possible to maintain the functionality of the robot. The grippers should also be kept as light as possible to ensure that the arm motors (for swing motion) provide enough torque to overcome the magnetic resistance associated with the clutches at shoulder joints.

Fig. 3. Simulation results of swing phase: effects of tail length and tail weight on maximum swing angle of robot arm.

3.2. Mechatronic system configuration Fig. 4 presents a 3D plot of the proposed TRBR-III, in which the parallelogram linkage (formed by the two arms, the body, and the ledge) provides an arm posture capable of maintaining a firm hold on the ledge during the swing phase. This design also minimizes the accumulation of excess angular momentum to reduce yaw rotation during the flight phase. We also employed a parallelogram linkage design in each arm to keep the grippers parallel to the ledge throughout the flight phase and thereby ensure a stable landing posture without the need for wrist actuators. This design allows placement of the arm and tail motors within the body section to reduce the moment of inertia of the arm and tail links. This design also reduces the mass of the robot and the torque required by the motor. Finally, this design reduces the clamping forces required during the swing phase. We employed electromagnetic clutches at the shoulder joints to provide stiffness during the swing phase and flight phase. Torque from the motor is delivered to the links through gears and belts, with the link motors positioned on the body so that the robot COM deviates only minimally from the vertical swing plane (i.e., roll rotation mentioned in Section 2) during the flight phase. The circuit boards for sensing and control are attached to the main body, and the battery packs and voltage regulator are used as tail mass to save space and reduce the total weight. Note that the actuators associated with the grippers are also placed within the body.

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 4. 3D plot of designed TRBR-III: (a) front view; (b) back view.

Fig. 5. Conceptual design of robotic gripper: (a) holding unit; (b) actuating unit.

3.3. Gripper design

3.4. Sensory and control configuration

The mechanism used for the gripper in the previous generation of TRBR devices [28] was prone to unexpected triggering associated with shaking that occurred during the swing phase. In designing the TRBRIII, we employed a quick release mechanism for the grippers to provide more robust switching action between hold and release modes. The proposed gripper design in Fig. 5 comprises a holding unit and a much more massive actuating unit, which was mounted within the body mechanism to reduce the mass of the arm. The gripper holding unit includes a moving component and a fixed component. The gripper closes when tension is applied to the moving component via a control string connected to the actuating unit. Thus, higher tension produces a tighter clamping force. When the tension in the string is released, the gripper is forced open through the release of elastic energy in the rubber band shown in Fig. 5(a). The actuating unit of the gripper includes a DC motor, a small servo, a control string, and a quick release mechanism. In holding mode, the spool of the quick release mechanism is attached to the fixed part by the servo. Connecting the spool to the shaft of the DC motor allows the control string to be wound up, thereby storing energy in the elastic band for a controlled release. This design module makes it possible to reduce the diameter of the quick release spool and in so doing increase the tension in the control string (i.e., greater clamping force) for a given motor torque. When the spool is detached from the assembled mechanism, the elastic energy stored in the rubber band rapidly unwinds the control string from the spool to release the gripper.

The robot posture control algorithms presented in this study were implemented using the STM32F407 Discovery kit (STMicroelectronics, ARM Cortex-M4 with FPU core, 1-Mbyte Flash, 192-Kbyte RAM, 168 MHz CPU), which was mounted in the body link. TB6612FNG (TOSHIBA) motor driver boards were used for voltage command control in the servo loops of joint motors and HCTL-2032-SC (AVAGO) decoding integrated circuits were used to read encoder signals. The sampling rate for the motor feedback control system was set at 1 kHz. The arm and motor were detached during the swing phase, which made it impossible to acquire information related to arm swing angle from motor encoder readings. We therefore installed GY-953 (AHRS module) IMU sensors to the body link to determine the orientations of the arms and body for posture feedback control at a sampling rate of 100 Hz. Resistive pressure sensors (Interlink, FSR 402) were attached to the grippers to measure clamping forces during the swing phase in order to assess the performance of the proposed gripper. Switching by the clutches was controlled using relays, and a Hall effect sensor was used to identify the homing position of the robot tail. All of the data obtained in the experiments were first stored in a 23LC1024 1 Mbit serial SRAM device (Microchip), and then transmitted to an SD card for analysis. 4. Flight dynamics of TRBR As mentioned in Section 2, we simplified the complex flight dynamics of the ricochetal brachiating robot as transverse (2D) flight motion

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

A more detailed derivation of the rotational dynamics can be found in the literature [23,25,28]. Our previous application of an open-loop tail swing strategy to adjust the midair body posture demonstrated that the dynamics simulated using this simplified two-link model are fairly close to the dynamics of the actual robot in experiments [28]. Thus, we adopted the flight dynamics presented in (10) and (11) in developing a model on which to base our flight posture control strategy. 5. Active compensation for robot posture in midair

Fig. 6. Dynamic model of proposed ricochetal brachiation robot during flight phase.

The yaw rotations introduced by the angular momentum during the flight phase can have a profound effect on the posture of the robot, which can lead to a decrease in the overall flight distance and/or skew the position of the grippers, thereby preventing a successful landing. In a previous study [28], it was shown that the tail of the robot could be rotated to alleviate yaw rotation. In this study, we implemented an active tail-assisted compensation strategy to adjust the body posture with the aim of maintaining a vertical body posture (i.e., 𝜃 b = 90°) throughout the flight. In the following, we first introduce the proposed posture control strategy based on rotational dynamics and then elucidate the effects of adjusting the body posture via simulations. 5.1. Midair posture controller

to facilitate the design of a posture control system. Even in this simplified configuration, the flight dynamics of the robot included two parts: translational dynamics and rotational dynamics. Translational dynamics was analyzed using the equation of projectile motion (2), and we present in the following a brief review of rotational dynamics [28]. The clutches at the shoulder joints are activated during the flight phase; therefore, the arms and body can be treated as a single unit, such that the rotational dynamics can be derived using a dynamic twolink model comprising an integrated arm-body link and a tail link, as shown in Fig. 6. The actual parameter values were obtained using the computer-aided design software. In Fig. 6, 𝜃 b refers to the angle between the center line of the robot body and the horizontal line, 𝜃 t is the angle between the extension line of the center of the tail and the horizontal line, and 𝜃 m is the angle between the center line of the robot body and the center line of the tail. 𝑚̄ 𝑏 = 0.9827 kg indicates the mass of the body and two arm links; mt = 0.3052 kg indicates the mass of the tail; mr = 1.2879 kg indicates the total mass of the robot; lb = 0.0589 m indicates the distance between the COM of the body link and the rotating shaft of the tail; lt = 0.0893 m indicates the length of the tail; Ib = 0.0015 kg·m2 indicates the moment of inertia of the lumped arm-body link; It = 1.923 × 10−4 kg·m2 indicates the moment of inertia of the tail link; and 𝜏 t indicates the torque applied to the tail. Let the position of the COM of the robot be the origin of the coor𝑇 dinate system with state variables 𝑥 = [𝜃𝑚 , 𝜃̇ 𝑚 ] = [𝑥1 , 𝑥2 ]𝑇 and control input 𝑢 = 𝜏𝑡 . The rotational dynamics based on the simplified two-link model can be represented as follows: 𝑥̇ = 𝑓 (𝑥) + 𝑔 (𝑥)𝑢 [ ] 𝑥2 2 2 𝑓 (𝑥) = 𝑄𝑅𝑆 𝑥2 +𝑆 𝐻0 ,

[ 𝑔 (𝑥 ) =

𝑇 (𝑄+𝑅)

0

]

𝑄+𝑅 𝑇

(10)

Angular momentum H0 resulting from the rotational dynamics of the robot can be expressed as ( ) ( ) 𝐻0 = 𝑀 − 𝐿 cos 𝜃𝑚 𝜃̇ 𝑏 + 𝑁 − 𝐿 cos 𝜃𝑚 𝜃̇ 𝑡 (11) where 𝜃𝑡 = 𝜋 − 𝜃𝑚 + 𝜃𝑏 . The other variables in (10) and (11) are summarized as 𝑀 = 𝐼𝑏 +

𝑚𝑡 𝑚̄ 𝑏 𝑙𝑏2

, 𝑚𝑡 + 𝑚̄ 𝑏 𝑄 = 𝑀 − 𝐿 cos 𝑥1 ,

𝑚𝑡 𝑚̄ 𝑏 𝑙𝑡2 𝑚 𝑚̄ 𝑙 𝑙 , 𝐿 = 𝑡 𝑏 𝑡 𝑏, 𝑚𝑡 + 𝑚̄ 𝑏 𝑚𝑡 + 𝑚̄ 𝑏 𝑅 = 𝑁 − 𝐿 cos 𝑥1 , 𝑆 = 𝐿 sin 𝑥1 ,

𝑁 = 𝐼𝑡 +

𝑇 = 𝑀𝑁 − 𝐿2 cos2 𝑥1

(12)

Consider the system output for the flight dynamics represented by Eq. (10) in the previous section:𝑦𝑐 = [𝜃𝑏 , 𝜃̇ 𝑚 ]𝑇 . The robot comprises multiple linkages that differ in the moment of inertia, and is equipped with feedback sensors with different sampling rates. Thus, the proposed system can be regarded as a single-input-two-output (SITO) system with fast as well as slow dynamics. This allowed us to develop a control structure based on successive loop closure (SLC) [33] in order to simplify the design of the underactuated control system to maintain body posture control. A control block diagram is presented in Fig. 7. The torque provided by the tail motor 𝜏 t is the control input; therefore, the system plant Probot can be expressed as follows: 𝑥̇ = 𝑓 (𝑥) + 𝑔 (𝑥)𝑢 𝑦 𝑐 = ℎ 𝑐 (𝑥 ) [ 𝐻 +𝑅𝑥 ] 0 2 ∫ 𝑄 + 𝜃 𝑏 (0 ) +𝑅 ℎ 𝑐 (𝑥 ) = 𝑥2

(13)

where 𝜃 b (0) denotes the body angle at the instant of leaping. For the sake of illustration, the differentiation of 𝜃 b in (13) gives the following expression: 𝜃̇ 𝑏 =

𝐻0 + 𝑅𝜃̇ 𝑚 . 𝑄+𝑅

(14)

The control system in Fig. 7 contains two control loops: an inner loop and an outer loop. The inner control loop (corresponding to the faster dynamics associated with the smaller moment of inertia of the tail) uses system input 𝜏 t and system output 𝜃̇ 𝑚 . The outer control loop system (corresponding to the slower dynamics associated with the larger moment of inertia of the body) uses system input 𝜃̇ 𝑚 and system output 𝜃 b . The objective of the SLC-based control system is to design controllers Cinner and Couter for the two control loops by which to adjust the angular velocity of the tail 𝜃̇ 𝑚 (or equivalently 𝜃̇ 𝑏 according to (14)) and maintain the body angle 𝜃 b at the reference value 𝜃𝑏𝑟 = 90◦ . The output of the outer loop controller Couter is 𝜃̇ 𝑏𝑟 , which is the reference angular velocity required for the robot body to assume the desired posture (𝜃 br ). The reference input trajectory for the inner loop control system, 𝜃̇ 𝑚𝑟 , can be derived using (14). Note that the sampling rate (1 kHz) of the inner loop system is 10x faster than the sampling rate of the outer loop system (100 Hz). We applied an averaging filter to obtain a smoother reference trajectory 𝜃̇ 𝑚𝑟 in order to avoid chattering effects due to interactions with the slower system dynamics and thereby enhance control performance. Here, we take into account the motor dynamics Pmotor related to voltage

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 7. Block diagram of proposed body posture control (BPC) system.

Fig. 8. Block diagram of tail regulation control system (i.e., without body posture control).

input Em and provided motor torque 𝜏 t (Fig. 7) to facilitate subsequent implementation. The dynamic equation can be represented as follows: 𝐸𝑚 = 𝑅𝑖𝑚 + 𝐾𝑚 𝜔𝑚 𝑑 𝜔𝑚 𝐽𝑚 = 𝐾𝑚 𝑖𝑚 − 𝑏𝜔𝑚 − 𝑐 ⋅ sgn(𝜔𝑚 ) 𝑑𝑡

(15)

where variables im and 𝜔m respectively refer to the current and speed of the motor. Details of the motor parameters in (15) are listed in Table 2. The numerical values were obtained using a least-squares based system identification procedure [35]. In this study, we adopted the control block diagram shown in Fig. 7 as the basic scheme for body posture compensation, which is abbreviated as BPC in the following discussion. To highlight the benefits of the proposed method, Fig. 8 presents a comparison control block diagram, which is tasked only with controlling the angle related to tail posture (𝜃 m ) during the flight phase. Angular momentum generated by free tail rotation can undermine the stability of flight posture results. Thus, our objective in formulating this control strategy (referred to as “tail regulation control” (TRC)) was to maintain a constant angle between the body and the tail, i.e., to eliminate any relative motion between the two links. In Fig. 8, 𝜃 mr represents the reference input, which is the angle of the tail at the instant of leaping and Cm is the position controller. 5.2. Simulation results As described in the previous sections, the robot should initially be under resonant excitation before proceeding into subsequent phases. The reference trajectory of the robot tail for swing excitation can be derived using the model-based kinetic energy optimization method in [28]. Based on the design parameters obtained in Section 3, we adopted a sinusoidal signal (35° at 1.47 Hz) as the tracking trajectory for the Table 2 Model parameters of DC motor in robot tail. Symbol

Meaning

Value

R Km Jm b c

Motor resistance Torque constant Rotational inertia Viscous friction coefficient Coulomb friction coefficient

2.5945 Ω 0.02689 N·m/A 4.5984 × 10−5 kg·m2 2.3903 × 10−4 N·m·s 0.00168 N·m

Fig. 9. Simulation results of robot swing excitation.

analysis of swing angle using simplified 2-DOF swing dynamics [27]. This was then used to determine the leap angle and timing required for ricochetal brachiation. Fig. 9 presents the simulation results of resonant excitation during the swing phase, where the blue solid line indicates the angle between the center line of the robot arm and the vertical line (i.e., swing angle 𝜃 as ), and the red dashed line indicates the angle between the center lines of the tail and body (𝜃 ts ). If the swing angles were between -90° and 90°, then the magnitude of 𝜃 max would be 90°. Applying this condition to Eq. (8), we obtain a theoretical value for the maximum transverse leap distance when the swing angle 𝜃 as = 35.25° (or equally the leaping angle 𝜃 j defined in Section 3). In this simulation, we selected a leap time of 5.59 s, as indicated by the red vertical line in Fig. 9. We employed flight dynamics (described in Section 4) in the performance assessment of the proposed robot posture compensation methods following transition to the flight phase. Motor voltage Em was limited to between -12 V and 12 V based on the specifications of the motor drivers. Fig. 10 (a) and (b) shows the output responses and control input under the TRC control scheme. Controller Cm is a proportionalintegral-derivative (PID) controller with the following parameter values: kp = 0.3, ki = 0.1, kd = 0.1. Our results show that after fine-tuning the parameters, the tail angle (𝜃 m ) effectively followed the reference command 𝜃 mr ; however, yaw rotation caused body angle (𝜃 b ) to deviate gradually from the target value 𝜃 br . In real-world implementations, this could compromise the landing posture, thereby preventing the robot from maintaining a stable hold on the target ledge. Disregarding these effects, let us assume that the robot is still able to hold onto the target ledge under this body posture. Transverse motion is calculated using the equation of projectile motion (2) associated with the COM of the robot. As indicated by the vertical red lines in the plots, the flight duration without body posture compensation (TRC) would be 0.14 s and the landing time would occur at 5.73 s. Fig. 10(c)–(f) presents the simulation results for the case using the proposed posture compensation method (BPC). In these figures, the

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 10. Simulation results obtained during flight phase indicating effects of robot posture control: (a) output responses obtained using tail regulation control only; (b) control input using tail regulation control only; (c) output responses using proposed body posture control (outer control loop); (d) zoomed-in plot of (c); (e) output responses using proposed body posture control (inner control loop); (f) control input using proposed body posture control.

faded results after landing are presented only as a reference, due to the fact that we did not address the issue of landing dynamics in this study. We adopted proportional-integral (PI) controllers for the inner and outer control loop systems. The design parameters were obtained using the Response Optimization Toolbox in Matlab using the following control specification settings: reference input = 90°, rising time = 0.05 s, settling time = 0.1 s, and undershoot = 10%. The pattern search algorithm returned the following parameter values: outer loop system (kp = 41.548, ki = 7.761) and the inner loop system (kp = 0.254, ki = 0.634). The plots in Fig. 10 show that body angle 𝜃 b followed the reference command within 0.1 s (5.59 s ~ 5.69 s), and the output response of the inner control loop system (𝜃 m ) reached a steady state in 0.08 s (5.59 s ~ 5.67 s). The faster response of the inner loop control system can be attributed to the characteristics of the SLC-based control scheme. Note that the flight time when using the BPC scheme was 0.153 s, which is 0.013 s longer than the flight time using the TRC scheme. Fig. 10 shows the performance improvement provided by the BPC scheme. Our objective was to compensate for deviations in body angle

within a very short time; therefore, it is not surprising that the control input became saturated within the first 0.02 s, as shown in Fig. 10(f). Furthermore, the piecewise continuous behaviors exhibited in the control voltage could potentially have negative effects on the performance in subsequent experiments. In plotting the movement trajectory of the left gripper, we employed the equation of projectile motion, treating the arms and body as a single unit during the flight phase. Our aim was to visualize the difference in performance between the two posture control methods, the results of which are presented in Fig. 11. The origin of the 2D coordinate system is defined as the location from which the robot releases its grippers. Red lines indicate the body postures, the solid line indicates the arm-body link, and the dashed line indicates the tail link. From this, we can see that maintaining a vertical body posture enhances flight performance (30.17 mm in height, 166 mm in distance), compared to performance when controlling only the tail (24.11 mm in height, 130 mm in distance). The vertical body posture also sets up the grippers in a position that is favorable to landing.

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 11. Simulations of robot flight trajectories: (a) without body posture control (tail regulation control); (b) with body posture control; blue line indicates trajectories of left gripper, red lines indicate simplified robot postures with 2 DOF. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6. Experiments and discussion

Fig. 12. Strategies for arm motion during flight phase: (upper plot) without arm swing; (lower plot) with arm swing.

In this study, extruded aluminum bars were used in constructing the experiment platform for experiments on transverse ricochet locomotion [27,28]. Based on the active tail control methods presented in Section 5, we implemented a number of flight posture compensation strategies to evaluate the effectiveness of the proposed scheme.

accordance with the law of the conservation of angular momentum, this arm rotation caused the robot to rotate counterclockwise, which had a detrimental effect on the angle of the body. 6.3. Transverse ricochet experiments

6.1. Experiment setup Swing excitation tests revealed that the maximum swing angle of the proposed robot was 𝜃 max = 65°. The discrepancy between this value and the ideal value derived in simulations (exceeding 80°) can be attributed to frictional resistance in the joints, a lack of stiffness in the bars holding the grippers, and a lack of stability in the grabbing motion due to the effects of shaking imposed by the swinging robot tail. Substituting parameter 𝜃 max = 65° into Eq. (8) changed the transverse flight distance xjd to 11.92 cm (under the condition that 𝜃 j = 37.51°), which is approximately 73% of the ideal value (16.29 cm subject to 𝜃 max = 80°). As shown in Fig. 2, the theoretical gap between adjacent bars can be derived as lg = xs + xj – ls = 6.02 cm, where the distance between the two arms (ls ) is set at 5.9 cm (Section 3). In light of the inevitable energy dissipation effects, we adopted a conservative setting for lg (5.5 cm) in subsequent experiments. 6.2. Integration of arm swing motion Under the same leap conditions, we incorporated two arm swing strategies within the proposed body posture control scheme in order to facilitate the adjustment of robot flight posture, improve the positioning of the grippers at landing, and extend the flight time and distance, as shown in Fig. 12. The strategy in the upper plot of Fig. 12 is meant to duplicate the arm posture at the instant of leaping for landing. The strategy in the lower plot of Fig. 12 involves rotating the arms in the direction toward the target bar, until they are parallel with the centerline of the robot body (𝜃 b = 90° ideally). In this study, all of the leaps were from the left bar to the right bar. 𝜃 PP indicates the size of the angle through which the robot must swing to reach the other side. The clockwise rotation of the robot arms brings the grippers closer to the target bar and in so doing extends the transverse distance traveled by the robot. However, in

Four sets of transverse ricochet experiments were performed to evaluate the performance of the proposed robot posture compensation methods: (1) tail regulation without arm swing (Exp 1); (2) body posture control without arm swing (Exp 2); (3) tail regulation with arm swing (Exp 3); and (4) body posture control with arm swing (Exp 4). In the experiments, images of the flight trajectories were captured using a high-speed camera (Photron FASTCAM SA2, 1000 Hz sampling). Performance was analyzed offline by applying a mark on the left gripper in the captured images. For the sake of convenience, let the leap position be the position in which the robot releases its grippers, and the landing position be the position in which the left gripper reaches the same elevation as the leap position. Based on these definitions, the flight distance refers to the difference in horizontal distance between the two positions, the flight time refers to the time that elapses as the robot moves between the two positions, and the flight height refers to the maximum height attained by the robot during projectile motion. The sliding and bouncing of the grippers after connecting with the target bar are beyond the scope of this study. 6.3.1. Swing excitation for leaping We applied a 1.47 Hz, 35° sinusoidal signal (as in the simulation) as a tail reference input for swing excitation aimed at the accumulation of kinetic energy for leaping. Fig. 13(a) presents the experimental results before transition to the flight phase, where the red dashed line indicates the tail angle 𝜃 ts (obtained from the tail motor encoder) and the solid blue line indicates the arm swing angle 𝜃 as (obtained from the IMU). Note that the validity of IMU data was confirmed via comparison with the data collected using the high-speed camera (green dashed line). As shown in the figure, the command for gripper release was sent at 4.536 s, which satisfied the following two conditions: (1) maximum swing angle must exceed 65°; (2) leap angle must exceed 37.51°. Fig. 13(b) clearly

C.-Y. Lin and Z.-H. Yang

Fig. 13. Transverse ricochet experiments (swing phase): (a) arm swing angle; (b) fluctuations in gripper force during gripper release process. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

shows that there was a variable time delay in the handholding force of both the left and right grippers, due to the effects of mechanical resistance. The pressure sensors are attached to the moving part of the grippers (lower claws) and the robot is rotating counterclockwise in releasing the grippers; therefore, the inertial forces have a more pronounced impact on the right gripper than on the left gripper. This gripper release process ended when the right gripper force disappeared at 4.58 s. In this study, we pre-compensated for the time delay during the phase transition (0.044 s) in order to achieve the required leap angle. Note that the position of both grippers at the moment of release is referred to as the leap position. 6.3.2. Analysis of active posture compensation methods In implementing the TRC scheme, we adopted the parameter values used in the simulations. In implementing the cascaded BPC scheme, model uncertainties and system nonlinearities meant that some of the parameters were too aggressive; i.e., they could lead to instability in real-world implementations. Thus, for the BPC scheme, we empirically adjusted the parameters as follows: outer loop control system (kp = 20) and inner loop control system (kp = 0.03, ki = 0.36, kd = 1.2). Calculations of angular momentum H0 took into account the law of conservation of angular momentum at the instant of leaping. The parallelogram mechanism made it relatively easy to apply angular momentum of the swing phase to simplify the calculation of H0 , based on the assumption that the robot body did not undergo any rotation (i.e., 𝜃 b = 0 and 𝜃̇ 𝑏 = 0)

Mechatronics 65 (2020) 102307

during the swing phase. PID controllers were used for arm motion control (regulation or tracking) using the following parameters: kp = 0.12, ki = 0.01, kd = 0.01. Figs. 14 and 15 present the results of the four experiments and Fig. 16 presents the flight trajectories of the left gripper, which were derived from snapshots recorded during the transverse ricochetal brachiation experiments. Statistical data are summarized in Table 3 and the simulation results are also included for comparison. Sim 1 indicates the results obtained using TRC and Sim 2 indicates the results obtained using BPC. The results obtained in Sim 1 and Sim 2 respectively correspond to the results obtained in Exp 1 and Exp 2 using a simplified two-link dynamic model with the arms and the body linked to form a single unit. The abbreviation ASM in the captions of the subfigures denotes arm swing motion. Only the experiments that included arm swing achieved successful two-arm brachiation leaps. All of the experiments conducted without arm swing lacked sufficient flight distance, which resulted in one-arm brachiation, under a designated gap distance of 5.5 cm. Our ultimate control goal was to maintain a vertical body posture for landing. The BPC scheme (Sim 2, Exp 2, and Exp 4) consistently outperformed the TRC scheme (Sim 1, Exp 1, and Exp 3) in terms of body angle during flight and at the point of landing. The conservative parameter settings employed in the BPC scheme resulted in phase lag effects in the output response and control input of the inner loop system, and these effects were more pronounced when arm swing was added (Exp 4). As shown in subfigures (c) and (d) of Figs. 14 and 15, the BPC scheme produced non-smooth zig-zag effects associated with the noise in IMU data transmission as well as unmodeled dynamics and inherent nonlinearities. The similarities between the simulation and experimental results demonstrate that our assumptions and modeling results are appropriate for the design and implementation of body posture control systems. Nonetheless, Exp 1 and Exp 2 failed to match the performance of the corresponding Sim 1 and Sim 2, due mainly to energy dissipation during the transition from swing phase to flight phase. Our results in Exp 3 and Exp 4 indicate that arm swing can be used to improve flight performance; however, it tends to alter the body angle at landing by introducing angular momentum associated with the arm swing. The results obtained in Exp 4 clearly demonstrate the benefits of using body posture control and the effectiveness of appropriate arm swing; however, considerable energy is required to coordinate the arms and tail. Fig. 16 presents the robot flight trajectories obtained from the highspeed camera, illustrating the complex dynamics associated with the flight phase and some of the mechatronic issues. The negative values appearing at the end of flight trajectories can be attributed to bending deformation of the target bars upon landing. Our objective in this study was to investigate flight performance using posture compensation methods; i.e., landing dynamics were beyond the scope of this study. The results in Fig. 16(b) present a flight trajectory that resembles a parabola. This result is the closest to the typical 2D trajectory seen in the motion of projectiles; however, it is slightly asymmetric. In the other three cases, all of the flight trajectories indicating the movements of the left gripper present non-monotonic ascending or descending changes throughout the flight, due to occasional collisions with the support bars and unexpected bouncing effects under the influence of body rotation. These impacts dissipate the kinetic energy stored during the swing phase, which decreases the height of the grippers during flight and thereby limits the flight distance. This is yet another reason for the lack of research into ricochetal brachiation robots. The results in Fig. 16(d) indicate that it is possible to compensate for the effects of impact through the simultaneous implementation of body posture control and an appropriate arm swing strategy. Snapshots of this complex motion are shown in Fig. 17. Nevertheless, advanced robot posture compensation methods and novel mechatronic design should be pursued further with the aim of reducing modeling uncertainties and mitigating the effects of sliding and bouncing.

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 14. Experimental results without arm swing (Exp 1 and Exp 2): (a) control input and output responses of TRC scheme; (B) output responses of BPC scheme (outer control loop); (c) output responses of BPC scheme (inner control loop); (d) control input of BPC scheme. Table 3 Evaluation of flight performance using various robot posture compensation strategies.

Arm swing motion Posture control strategy Flight distance (mm) Maximum flight height (mm) Landing angle (𝜃 b ) (deg) Flight time (sec) ∗

Sim 1

Sim 2

Exp 1

Exp 2

Exp 3

Exp 4

N/A∗ TRC 130 24.11 96.85 0.14

N/A∗ BPC 166.6 30.17 90.06 0.153

No TRC 46 4.25 101.4 0.122

No BPC 90.33 16.17 89.93 0.125

Yes TRC 107.8 6.538 111.6 0.135

Yes BPC 148.3 18.33 91.84 0.140

not applicable for simplified two-link flight dynamics

6.4. Comparison of TRBRs Table 4 lists the design features of each generation of TRBR as well as the key improvements: (1) TRBR I was the first prototype; (2) TRBR II and TRBR III include a switchable shoulder stiffness scheme as well as a procedure by which to optimize tail swing for more efficient resonant excitation; (3) TRBR II and TRBR III include custom-made grippers mimicking the grip and release actions of humans to improve locomotive phase transition; (4) TRBR III includes an arm-and-tail coordination motion control strategy to control the angle of the robot body and improve overall flight distance. In this study, we created a performance index to evaluate the differences between the TRBRs from the perspective of energy conversion efficiency. Index values are derived as follows: 𝑑𝑓 𝜂𝑓 = (16) 𝐸𝑠

where df represents the actual flight distance and Es represents the maximum kinetic energy stored during the swing phase. To simplify analysis, we assume that the energy consumed in arm posture realization is negligible and treat the robot as a single pendulum with all of the components lumped together as a point mass at the end of the pendulum, in which the length of the pendulum is equal to the original length of the robot arm la and the end mass is equal to the mass of the entire robot mr . Under the ideal situation where 𝜃 max = 90° and there is no energy loss, the ideal value of the maximum kinetic energy for swing motion Es can be approximated by calculating the potential energy from the law of energy conversion as follows: 𝐸𝑠 ≈ 𝑚𝑟 𝑔 𝑙𝑎 .

(17)

A larger 𝜂 f value indicates superior energy utilization and less energy dissipation during ricochetal brachiation, and vice versa. The results in Table 4 indicate that the TRBR presented in this study achieved energy conversion efficiency at least 1.5 times that of the two previous generations of TRBR. This clearly demonstrates the efficacy of the proposed robot design and mid-air posture compensation strategies.

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 15. Experimental results with arm swing (Exp 3 and Exp 4): (a) control input and output responses of TRC scheme; (B) output responses of BPC scheme (outer control loop); (c) output responses of BPC scheme (inner control loop); (d) control input of BPC scheme.

Fig. 16. Experimental results of transverse flight trajectories: (a) TRC scheme without arm swing motion; (b) BPC scheme without arm swing motion; (c) TRC scheme with arm swing motion; (c) BPC scheme with arm swing motion; blue lines indicate trajectories of left gripper, and red lines indicate postures of body and tail links. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

C.-Y. Lin and Z.-H. Yang

Mechatronics 65 (2020) 102307

Fig. 17. Snapshot sequence of transverse ricochectal brachiation experiment using proposed body posture control and arm swing motion strategy (Exp 4); red lines indicate postures of body and tail links.

Table 4 Design features of TRBR generations.

No. of actuations Shoulder stiffness Humanoid gripping action Gripper weight Robot parameter optimization Swing parameter optimization Operation mode Swing Time Mid-air arm swing motion Mid-air posture control Maximum flight distance Performance index (𝜂 f )

TRBR-I [27]

TRBR-II [28]

TRBR-III (this study)

5 Fixed No Heavy No No Manual ≈ 11 s No No 10 cm 6.796 cm/J

4 Switchable Yes Light No Yes Autonomous 5.8 s No Open loop control 14.6 cm 7.783 cm/J

7 Switchable Yes Light Yes Yes Autonomous 4.53 s Yes Closed loop control 14.8 cm 13.017 cm/J

7. Conclusions This paper presents a robot based on the arm-body-tail configuration for transverse ricochetal brachiation. We also present in-depth analysis of posture compensation methods and their effects on flight performance. We formulated an optimization problem aimed at maximizing the flight distance under given constraints with the aim of identifying the robot design parameters that are critical to flight performance. We employed shoulder joints with switchable stiffness as well as custommade grippers and an optimized tail swing excitation procedure to fa-

cilitate phase transition and reduce the associated dissipation of energy. Simulations using the proposed body posture control scheme were conducted to analyze the effects of yaw rotation during the flight phase and assess the effectiveness of body posture control methods in improving performance. Experimental results demonstrate the efficacy of the proposed control strategy involving tail swing and arm swing to maintain the angle of the robot during flight. Future research will focus on the development of grippers embedded with damping components to deal with uncertain impact dynamics upon landing. New robotic locomotion styles incorporating the coordinated control of separate two-

C.-Y. Lin and Z.-H. Yang

arm motion and tail swing could be also developed to extend leap distance. Declaration of Competing Interest None. Acknowledgment This work was supported by the Ministry of Science and Technology, Taiwan, under grant number MOST-105-2628-E-011-004 and partially supported by the “Center for Cyber-physical System Innovation” from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. References [1] Fukuda T, Hosokai H, Kondo Y. Brachiation type of mobile robot. In: Proceedings of the IEEE international conference on robotics; Jun. 1991. p. 915–20. [2] Nakanishi J, Vijayakumar S. Exploiting passive dynamics with variable Stiffness actuation in robot brachiation. In: Proceedings of the robotics science and system, Sydney, Australia; Jul. 2012. [3] Stöckli F, Shea K. Automated synthesis of passive dynamic brachiating robots using a simulation-driven graph grammar method. ASME J Mech 2017;139(9):092301–9. [4] Nakanishi J, Fukuda T, Koditschek DE. Preliminary studies of a second generation brachiation robot controller. In: Proceedings of the IEEE international conference on robotics and automation; Apr. 1997. p. 2050–6. [5] Saito F, Fukuda T, Arai F. Swing and locomotion control for two-link brachiation robot. In: Proceedings of the IEEE international conference on robotics and automation; May 1993. p. 719–24. [6] Bertram JE, Ruina A, Cannon CE, Chang YH, Coleman MJ. A point-mass model of gibbon locomotion. J Exp Biol 1999;202(19):2609–17. [7] Fukuda T, Saito F, Arai F, Kosuge K. A study on the brachiation type of mobile robot. Trans Jpn Soc Mech Eng Ser C 1991;57(541):2930–7. [8] Nakanishi J, Fukuda T, Koditschek DE. A hybrid swing up controller for a two-link brachiating robot. In: Proceedings of the IEEE/ASME international conference on advance intelligent mechatronics; Sept. 1999. p. 549–54. [9] Oliveira VM De, WF Lages. Comparison between two actuation schemes for underactuated brachiation robots. In: Proceedings of the IEEE/ASME international conference on advance intelligent mechatronics, Zurich, Switzerland; Sept. 2007. [10] Zhao Y, Cheng H, Zhang X. Swing control for two-link brathiation robot based on SMC. In: Proceedings of the IEEE conference on Chinese control decision; Jul. 2008. p. 1704–8. [11] Tashakori S, Vossoughi G, Yazdi EA. Geometric control of the brachiation robot using controlled Lagrangians method. In: Proceedings of the RSI/ISM international conference on robotics and mechatronics; Oct. 2014. p. 706–10. [12] Ebrahimi K, Namvar M. Port-Hamiltonian control of a brachiating robot via generalized canonical transformations. In: Proceedings of the American control conference; Jul. 2016. p. 3026–31. [13] Nguyen KD, Liu D. Robust control of a brachiating robot. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems; Sept. 2017. p. 6555–60. [14] Hasegawa Y, Fukuda T, Shimojima K. Self-scaling reinforcement learning for fuzzy logic controller-applications to motion control of two-link brachiation robot. IEEE Trans Ind Electron 1999;46(6):1123–31. [15] Kajima H, Hasegawa Y, Doi M, Fukuda T. Energy-based swing-back control for continuous brachiation of a multilocomotion robot. Int J Intell Syst 2006;21(9):1025–43. [16] Meghdari A, Lavasani SMH, Norouzi M, Mousavi MSR. Minimum control effort trajectory planning and tracking of the CEDRA brachiation robot. Robotica 2013;31(7):1119–29. [17] Nakanishi J, Fukuda T, Koditschek DE. Experimental implementation of “target dynamics” controller on a two-link brachiating robot. In: Proceedings of the IEEE international conference on robotics and automation; May 1998. p. 787–92. [18] Nakanishi J, Fukuda T, Koditschek DE. A brachiating robot controller. IEEE Trans Robot Autom 2000;16(2):109–23.

Mechatronics 65 (2020) 102307 [19] Yamakawa Y, Ataka Y, Ishikawa M. Development of a brachiation robot with hook-shaped end effectors and realization of brachiation motion with a simple strategy. In: Proceedings of the IEEE international conference on robotics and biomimetics; Dec. 2016. p. 737–42. [20] Nakanishi J, Fukuda T, Koditschek DE. Brachiation on a ladder with irregular intervals. In: Proceedings of the IEEE international conference on robotics and automation; May 1999. p. 2717–22. [21] Kuo CT, Li WY, Wang YH, Lin PC. Dynamic modeling analysis of a spider monkey robot. In: Proceedings of the international conference on advanced robotics and intelligent systems; Sept. 2017. p. 99. [22] Li WY, Wang YH, Kuo CT, Lin PC. Design and implementation of a spider monkey robot. In: Proceedings of the international conference on advanced robotics and intelligent systems; Sept. 2017. p. 62. [23] Nakanishi J, Fukuda T. A leaping maneuvre for a brachiating robot. In: Proceedings of the IEEE international conference on robotics and automation; Apr. 2000. p. 2822–7. [24] Gomes MW, Ruina AL. A five-link 2D brachiating ape model with life-like zero-energy-cost motions. J. Theor. Biol. 2005;237(3):265–78. [25] Wan D, Cheng H, Ji G, Wang S. Non-horizontal ricochetal brachiation motion planning and control for two-link bio-primate robot. In: Proceedings of the IEEE international conference on robotics and biomimetics; Dec. 2015. p. 19–24. [26] Cheng HT, Wan DK, Hao LN. Ricochetal brachiation motion planning and control for two-link bio-primate Robot. J Northeast Univ Nat Sci 2017;38(2):168–73. [27] Lin CY, Shiu SJ, Yang ZH, Chen RS. Design and swing strategy of a bio-inspired robot capable of transverse ricochetal brachiation. In: Proceedings of the IEEE international conference on mechatronics and automation; Aug. 2017. p. 943–8. [28] Yang ZH, Lin CY. Experimental investigation on flying motion of transverse brachiation robot. In: Proceedings of the IEEE/ASME international conference on advanced intelligent mechatronics; Jul. 2018. p. 1402–7. [29] Wu W, Huang M, Gu X. Underactuated control of a bionic-ape robot based on the energy pumping method and big damping condition turn-back angle feedback. Robot Auton Syst 2018;100:119–31. [30] Hodgins J, Raibert MH. Biped gymnastics. Int J Robot Res 1990;9(2):115–28. [31] Xin X, Liu Y. Control design and analysis for underactuated robotic systems. London: Springer; 2014. [32] Zhao J, Zhao T, Xi N, Mutka MW, Xiao L. MSU tailbot: controlling aerial maneuver of a miniature-tailed jumping robot. IEEE/ASME Trans Mechatron 2015;20(6):2903–14. [33] Beard RW, McLain TW. Small unmanned aircraft : theory and practice. Princeton University Press; 2012. [34] Wang MJ, Wang EMY, Lin YC. The establishment of anthropometric database in Taiwan. Hum Factors Ergon Soc Annu Meet Proc 2000;44(38):696–8. [35] Morozovsky N. Motor modeling and identification for fun and profit [PDF]. 2012. Available from: http://renaissance.ucsd.edu/courses/mae143c/Dyno_Lecture.pdf. Chi-Ying Lin received the B.S. and M.S. degrees from National Taiwan University, Taiwan in 1999 and 2001, respectively, and the Ph.D. degree from University of California, Los Angeles in 2008, all in mechanical Engineering. He is currently an Associate Professor in the Department of Mechanical Engineering at National Taiwan University of Science and Technology, Taiwan. His recent research interests include (1) design, modeling, and locomotion control of bio-inspired brachiating robots; (2) active vibration control of smart flexible structures using piezoelectric electrode configuration techniques; and (3) development of novel robotic systems integrated with the techniques of visual servoing and force feedback.

Zong-Han Yang received the B.S. and M.S. degrees from National Taiwan University of Science and Technology, Taiwan in 2016 and 2018, all in mechanical engineering. He is currently an Engineer in Yulon Motor CO., LTD. His research interests include mechatronic design of bio-inspired robots and intelligent autonomous vehicles.