## Abstract

Based on the cat-righting reflex, this paper presents two reorientation maneuvers for legged robots that can produce roll and pitch reorientation during free fall. In order to better describe and plan these maneuvers, two separate, but equivalent, theoretical frameworks that describe the kinematic and dynamic behavior of free-floating articulated architectures are explored and developed. A nine-degree-of-freedom quadruped robot architecture is then presented and used to demonstrate the proposed maneuvers. Finally, kinematic and dynamic simulations of this architecture are performed. The results validate the presented theoretical framework and demonstrate that both roll and pitch reorientations are obtained through the application of the presented maneuvers.

## 1 Introduction

Certain animals such as cats or lizards are capable of reorienting their bodies during a fall to ensure that they always land on their feet. Among the first to mathematically model this motion were Kane and Scher [1] with their two-cylinder cat, which demonstrated how cats were able to attain $180deg$ rotations about the roll axis. Other analyses of the mid-air reorientation maneuvers followed, including further work by Kane and Scher [2] on human reorientation, as well as papers studying the motion in other animals [3–5] and in certain sports, such as trampoline or diving [6,7]. Another approach explored the use of gauge theory in modeling the free-falling cat problem [8].

Along with a theoretical interest in reorientation maneuvers, interest in potential applications of the motion is also present, examples of which include satellite control using internal motions or reaction wheels [9–11] and effector control for robot arms in space [12–14], to name a few. A review on space robotics control is presented in Ref. [15]. More recent research on the topic focuses on the application of the cat-righting motion to terrestrial robots, the interest being impact mitigation during falls. This interest led to two separate ideas being developed, namely (1) reorientation using an added actuated appendage, inspired by reptilian reorientation maneuvers by means of a tail, and (2) reorientation using closed-loop torso and leg movements^{1} inspired by Kane and Scher’s self-righting cat model. As far as reptilian-based methods are concerned, Jusufi et al. [16] were among the first to present a reorientation technique by which the addition of an appendage could allow for pitch axis rotation, which they demonstrated using a reptile-inspired prototype. This method was then improved upon through testing on wheeled robots in a subsequent series of papers [4,17,18]. Further work on applied reptilian self-righting motions includes the use of an actuated tail for landing position control in jumping robots [19,20]. Regarding closed-loop motion inspired by the self-righting motion of cats, notable contributions include the application of closed-loop maneuvers for the orientation control of free-floating planar manipulators [21] as well as the series of papers studying the impact of the amplitude scaling of the internal motions on the resulting motion in terms of reorientation amplitude and energy and force efficiency [22,23].

Although much successful research pertaining to applied reorientation maneuvers has been done since Kane and Scher first published their self-righting cat model, most of the related work is limited to reorientation about a single axis. For the reptilian-inspired reorientation maneuvers, this is somewhat of an intrinsic limitation since the use of a single revolute joint at the base of the added appendage only allows for reorientation about either the pitch or the yaw axes. Although the number of degrees of freedom (DoF) between the tail and the robot could be increased, the resulting reorientation is likely to be limited without sacrificing some of the method’s simplicity. An example of this is work done by Chang-Siu et al. [24], in which multi-axis reorientation of a reptilian style robot is achieved with a two-DoF tail at the cost of a more sophisticated control system which must consider workspace limitations and avoid singularities.

However, in the case of the cat-inspired maneuvers, which are generally more elaborate and complex than the tail-based maneuvers, there is no physical limitation as to why the resulting reorientation should be limited to one axis of movement. Garant and Gosselin [25] address this by developing a three DoF articulated robot capable of completing two separate, closed-loop maneuvers which allow for reorientation about the roll and the pitch axes during free fall. Given that any given rotation in three-dimensional space can be described as a linear combination of rotations about any two distinct axes [26,27], Garant et al.’s robot is thus capable of producing any desired reorientation provided that the fall lasts long enough. Nevertheless, notably for shorter falls, reorientation capabilities along the robot’s roll and pitch axes would allow for much more efficient maneuvers when it comes to controlling the desired landing configuration. Indeed, the direction in which the robot faces when landing is not as important as having it land in a configuration in which its main body axis is parallel to the plane of the landing surface (generally the horizontal plane) as well as right side up.

In this paper, a nine-DoF articular architecture capable of reorientation about the roll and the pitch axes is presented with the aim of addressing the aforementioned limitations present in existing architectures. Notably, this architecture builds upon the work presented in Ref. [25] in order to obtain high-amplitude reorientation maneuvers about both horizontal axes ideal for controlling its landing configuration. Furthermore, the presented architecture, built as a generalized four-legged robot, illustrates the cat-inspired maneuvers in a conceptually functional context and demonstrates the minimal articular requirements to achieve these maneuvers. Comparisons between the proposed architecture and well-known legged robots are drawn later in this paper to illustrate this point. The structure of the paper is as follows: Sec. 2 briefly presents the theoretical framework used to describe the motion of free-floating robots and plan the proposed maneuvers, Sec. 3 presents the proposed architecture and describes the maneuvers used for reorientation, and Sec. 4 presents the simulated results that demonstrate the reorientation capabilities of the proposed architecture and validate the theoretical framework. Finally, concluding remarks are given in Sec. 5.

## 2 Kinematic and Dynamic Analysis of Free-Floating Robots

In this section, the forward kinematics and dynamics of a general free-floating articulated structure, illustrated in Fig. 1, is solved with the aim of obtaining mathematical models that can be used to plan and validate the maneuvers presented in the following sections of this paper. Since free-floating robots lack a fixed attachment, standard methods for solving these problems cannot be directly applied due to a lack of constraints. Dubowsky and Papadopoulos [13] address this problem by using the virtual manipulator approach. In this approach, a fixed reference frame is set at the robot’s center of mass and is then linked up to the end effector using a virtual articulated chain for which the forward kinematics problem can be solved sequentially. Using this method, the end effector kinematics can be fully defined, from which the kinematics of the remaining links in the chain can also be obtained.

In the context of reorientation maneuvers, the goal is to control the robot’s orientation, which can be described using the orientation of any of the links in the chain, in this case, the first link in the chain, referred to as the base link. Similar to Papadopoulos and Dubowsky’s virtual manipulator approach, a fixed reference frame is set at the robot’s center of mass and is linked to the base link by means of a virtual, six-DoF joint. Three of these DoFs describe the orientation of the base link with respect to the fixed reference frame and the other three describe the components of the position vector of the base link’s center of mass with respect to the fixed reference frame. Once these six parameters are known, the forward kinematics and dynamics problems can be applied to free-floating manipulators. In order to define these six DoFs, additional constraints must be considered. Sections 2.1 and 2.2 show how force and momentum conservation principles can be used to define the required constraints.

### 2.1 Augmented Dynamic Model.

**M**

_{aug}is the robot’s generalized inertia matrix derived considering the augmented articular parameters, and $M\u02d9aug$ is its derivative with respect to time,

*T*is the total kinetic energy of the manipulator and the augmented articular position, velocity, and acceleration vectors are defined as

**r**

_{b}, $r\u02d9b$, and $r\xa8b$ are the base link’s position, velocity, and acceleration vectors, and $\theta b$, $\omega b$, and $\omega \u02d9b$ are the base link’s orientation, angular velocity, and angular acceleration vectors (all with respect to the fixed reference frame). It is worth noting that Eq. (1) does not depend on the robot’s potential energy because it does not affect the robot’s internal dynamics in the case of free-floating and free-falling robots.

**r**

_{b}from the robot’s center of mass. External forces must also be known or modeled. For free-falling applications, friction forces are generally the main external forces to consider, but are often negligible for short falls. In such cases, $\tau ext$ from Eq. (1) can be replaced with the zero vector of appropriate dimension. The initial values of

**M**

_{aug}, $M\u02d9aug$, and $\u2202T/\u2202\theta aug$ can then be obtained, leaving only $r\xa8b$, $\omega \u02d9b$, and $\tau $ as unknowns in Eq. (1). Moreover, by defining

**M**

_{ij}and

**C**

_{i}are submatrices of

**M**

_{aug}and

**C**, respectively. Since it no longer depends on $\tau $, Eq. (6) can be solved for $r\xa8b$ and $\omega \u02d9b$, which are then inserted in Eq. (5) to obtain $\tau $. $r\xa8b$ and $\omega \u02d9b$ are then integrated in order to obtain $r\u02d9b$ and $\omega b$, which are used to determine the base link’s orientation for the following time-step, and

**r**

_{b}is obtained from the updated position of the robot’s center of mass. The values of

**M**

_{aug}, $M\u02d9aug$, and $\u2202T/\u2202\theta aug$ for this time-step can then be determined, after which Eq. (6) can be solved and so on. By repeating this process for each point in the prescribed trajectory, it is possible to fully describe the kinematics and dynamics of a free-floating robot.

### 2.2 Model Based on the Conservation of Momentum.

Another approach that can be used to describe the kinematic and dynamic behavior of free-floating robots is similar to the one presented in Ref. [25]. In this approach, the forward kinematic and dynamic problems are dealt with separately. Indeed, the laws of conservation of linear and angular momentum are first used to complete the missing constraints in the kinematics problem and the obtained solution is then used to round out the dynamics problem.

*j*indicates the link,

*m*

_{j}its mass,

**r**

_{j}its position vector with respect to the robot’s center of mass, $r\u02d9j$ its velocity vector,

**I**

_{j}its inertia matrix, $\omega j$ its angular velocity vector,

**p**

_{j0}its initial linear momentum, and

**l**

_{j0}its initial angular momentum. By assuming a stationary center of mass as well as by using the physical constraints that connect each link in the chain, it is possible to express all the kinematic parameters of each link as a function of the kinematic parameters of the link that precedes it in the chain, namely

**A**and

**B**of dimension 6 × 6 and 6 ×

*k*, respectively. It is worth noting that to obtain Eq. (12), skew-symmetric matrices are used to represent the cross products that are present in Eq. (8) in order to facilitate term factorization [29]. One then obtains

### 2.3 Comparison.

Although both approaches successfully describe the kinematics and dynamics of a free-floating robot in the given conditions, it is important to note their differences in order to determine when one is to be favored over the other, both when describing the dynamics of any free-floating articulated architecture and planning articular trajectories for any desired maneuver. The main points on which these approaches differ are in terms of complexity and versatility. Indeed, the model based on momentum conservation provides a more straightforward approach in which the kinematics and dynamics can be solved separately and analytically, while the augmented dynamic model requires a numerical approach to solve the kinematics and dynamics of the robot. However, the augmented dynamics model provides a greater level of flexibility since the torque constraints added to the augmented dynamic model can be used to represent external forces, such as drag forces, acting upon the free-floating robot. This is not possible in the momentum conservation-based approach since momentum is only conserved in the absence of external forces. It is worth noting that both methods can describe motions with nonzero initial linear and angular momentum, assuming that these initial conditions are known.

## 3 Proposed Architecture and Reorientation Maneuvers

The proposed nine-DoF robot architecture, shown in Fig. 2, is composed of a square base to which four two-DoF, RR appendages are attached. The architecture of these appendages is inspired from the robot presented in Ref. [25], with, referring to the configuration shown at the top of Fig. 2, the first DoF allowing a rotation of the appendage about the *y*-axis (joints R5–R8) and the second DoF allowing rotations of the end piece about the *x*-axis (joints R9–R12). The remaining DoF controls the four rotary joints (R1–R4) connecting each appendage to the base, which are coupled by a mechanism that is not represented in Fig. 2. These serve solely to switch between the configurations.

The reorientation capabilities of the proposed robot architecture can be divided into two separate closed-loop trajectories, shown in Fig. 3. Closed-loop trajectories are preferred to open-loop ones because they ensure that the initial and final configurations of the robot are the same. Based on the conservation of angular momentum, the presented trajectories both reorient the robot along the same axis but differ in the amplitude of reorientation.^{2}

For the first maneuver (coarse maneuver), as shown on the left-hand side of Fig. 3, all four end pieces are simultaneously turned a full rotation in the same direction, which causes the robot to rotate about the same axis in the opposite direction. This maneuver can be repeated as many times as required to reorient the falling robot, but cannot provide reorientation of smaller amplitude than the minimum determined by the ratio of inertia between the four end pieces and the remainder of the links due to the closed-loop nature of the maneuver.

The second maneuver (fine maneuver), inspired by Garant and Gosselin [25] and shown on the right-hand side of Fig. 3, is more elaborate, but more flexible than the first one. Indeed, it can be used to obtain rotations of a smaller amplitude than the coarse maneuver. The first step in the fine maneuver is to fold up the appendages of the robot. Doing so increases the inertia of the main body and appendages to a value much larger than that of the end pieces about the end pieces’ axis of rotation, meaning that the end pieces can be rotated with little induced rotation of the remaining links. The end pieces are then wound up to a certain angle in the direction opposite to the one that induces the desired rotation. After unfolding the appendages, the end pieces can be returned to their initial position, inducing the desired rotation of the robot. For end piece rotations of less than 360 deg, this induced rotation can be of smaller amplitude than the minimal rotation induced by the coarse trajectory, while still maintaining the closed-loop nature of the maneuver due to the increased body and appendage inertias within the intermediate configurations (second and third images in Fig. 3). It is also worth noting that the folding angle of the appendages (controlled by R5–R8) can also be a parameter used to adapt the fine maneuver depending on the required rotation.

With these maneuvers and the ability to change configurations, the proposed architecture is capable of reorienting itself about the roll and pitch axes and, therefore, can control its landing configuration. Furthermore, with enough time, any desired three-dimensional rotation of the free-falling robot can be obtained by sequencing these maneuvers and configuration changes. The exact sequence of maneuvers and configuration changes required for a given desired reorientation can be determined with the theoretical frameworks presented in the previous section.

It is important to mention that the proposed architecture is built as a simplified quadruped robot. This is done in order to show how the earlier presented maneuvers can be completed by certain existing legged robot architectures with little to no modification. Indeed, the coarse maneuver can be completed by robots whose appendages have at least two R joints allowing yaw and roll rotations (R1 and R5 in Fig. 2). Examples of quadruped robots that meet these requirements are the Tekken robot [30–32], made to walk on difficult and irregular terrain, and the walking and crawling ALoF robot [33]. Furthermore, some highly complex robots, such as certain humanoid robots, are also capable of such reorientation maneuvers, an example of which is the BHR-5 robot [34]. However, it should be noted that some quadruped robots such as Boston Dynamic’s BigDog [35] and the MIT Cheetah [36] do not have yaw control of their appendages, meaning that they cannot achieve both pitch and roll reorientations without additional R joints on their appendages or a restructuring of their appendage architecture.

## 4 Simulated Results and Discussion

In order to validate the theoretical models and the maneuvers presented, the robot architecture shown in Fig. 2 is modeled in a *Siemens NX* motion simulation (Fig. 4), and both the reorientation trajectories are reproduced (Fig. 5). Mass and inertia values of each link are given in Table 1. For all presented simulations, it is assumed that the robot has no initial momentum. In the context of this paper, this assumption is justified as initial linear momentum is of no consequence for reorientation maneuvers and initial angular momentum only affects trajectory planning and does not affect the feasibility of the motions. However, in a more practical context, nonzero initial angular momentum could be measured using certain sensors, such as gyroscopes, and then factored in during the trajectory planning phase. Furthermore, external forces, such as friction forces, are not considered in these simulations. Indeed, most terrestrial applications of reorientation maneuvers are likely to occur for short falls during which falling speeds do not reach high enough values for aerodynamic friction forces to realistically affect the maneuvers. Moreover, given that the model based on the conservation of momentum cannot appropriately model cases in which external forces are present and impactful, the simulations must be done in the absence of such forces in order to validate both presented models in the same context.

Inertia (×10^{−4} kg m^{2})^{a} | ||||
---|---|---|---|---|

Link | Mass (kg) | Roll | Pitch | Yaw |

Base | 0.0646 | 0.578 | 0.578 | 1.102 |

L1–L4 | 0.0268 | 0.0748 | 0.133 | 0.115 |

L5, L8 | 0.0562 | 0.153 | 0.661 | 0.774 |

L6, L7 | 0.0511 | 0.149 | 0.518 | 0.630 |

L9–L12 | 0.0266 | 0.0891 | 0.0828 | 0.0107 |

EC-i 40 | 0.170 | 0.238 | 0.296 | 0.238 |

RE-max 17 | 0.026 | 0.00944 | 0.704 | 0.697 |

Inertia (×10^{−4} kg m^{2})^{a} | ||||
---|---|---|---|---|

Link | Mass (kg) | Roll | Pitch | Yaw |

Base | 0.0646 | 0.578 | 0.578 | 1.102 |

L1–L4 | 0.0268 | 0.0748 | 0.133 | 0.115 |

L5, L8 | 0.0562 | 0.153 | 0.661 | 0.774 |

L6, L7 | 0.0511 | 0.149 | 0.518 | 0.630 |

L9–L12 | 0.0266 | 0.0891 | 0.0828 | 0.0107 |

EC-i 40 | 0.170 | 0.238 | 0.296 | 0.238 |

RE-max 17 | 0.026 | 0.00944 | 0.704 | 0.697 |

Note: Mass and inertia electronics to operate the motors are not included in these values.

Inertia values are given with respect to each link’s center of mass.

During the simulation, the required torques and resulting base link (square base in this case) kinematics are monitored and compared to the corresponding theoretical values for specific trajectory amplitudes. These results for the roll configuration are shown in Figs. 6–9. The results from the simulation, the augmented dynamics and the momentum conservation-based model are essentially the same (superimposed curves). Actuator torques for joints R1–R4 are not monitored as these joints are locked during the maneuvers. As it can be seen in these figures, the dynamic and kinematic behavior of the robot is accurately modeled by both the presented theoretical methods for the roll configuration (the different curves can hardly be distinguished on the graphs). Nearly identical results are obtained for the pitch configuration, from which the same conclusions can be drawn. These results are omitted to avoid redundancy. Therefore, the simulation results validate the presented theoretical models.

In order to verify and characterize the reorientation obtained from the proposed maneuvers, the base link’s orientation is also monitored during the motion simulation and is shown in Figs. 10 and 11. From these figures, it is possible to note that the rotation of the base link is induced about the desired axis in all cases, validating the reorientation capabilities of the proposed maneuvers. It is also important to note that, for the chosen trajectory parameters, the fine maneuver yields a rotation of smaller amplitude than the coarse maneuver, which validates that, for certain trajectory parameters, the fine maneuver can induce rotations of smaller amplitude than the coarse maneuver. This also supports the idea of using these maneuvers sequentially to obtain the desired rotation of the base link, should the coarse maneuver not provide sufficient precision.

Even though it is not the primary aim of the presented work, it is important to determine the feasibility of these motions. Throughout both maneuvers, for the chosen time frame, the torque requirements reach a maximum of 4.88 Nm for joints R5–R8 and of 0.345 Nm for joints R9–R12. With a gearbox ratio of 33:1, the Maxon *EC-i 40* motor is capable of generating torques larger than 0.500 Nm in bursts while still providing the required motor speeds. This is not the case with the *RE-max 17* motor, which cannot meet both torque and speed requirements to complete the desired motions. Nevertheless, this motor can be replaced with a more powerful motor, such as the Maxon *EC-max 22*, which, with a gearbox ratio of 5.8:1, is capable of providing torques larger than 0.400 Nm in bursts while generating the desired motor speeds.^{3} Therefore, it is feasible that the proposed reorientation maneuvers be completed in the chosen time frame. Moreover, the simulation results show that a reorientation of 21 deg can be attained within 0.06 s with the coarse maneuver and that finer reorientations can be achieved within 0.16 s with the fine maneuver. This suggests that a full $180deg$ reorientation could be achieved roughly within the time of a 2 m fall (0.65 s). Indeed, by completing the coarse maneuver eight successive times, inducing a $168deg$ reorientation, followed by the fine maneuver to induce a $12deg$ reorientation, a $180deg$ rotation of the base link can be achieved in 0.65 s. If the reorientation of the fine maneuver is not needed, the coarse maneuver could be completed seven times within the time of a 1 m fall (0.45 s) for a total reorientation of 147 deg.^{4} Indeed, it can reasonably be assumed that legged robots could handle a certain orientation error at landing.

Although the obtained simulated results support the theoretical models and the reorientation maneuvers proposed in this paper, there are a few limitations to the work done in this paper that should be mentioned. First, no experimental validation of the presented methods is given, limiting the conclusions that can be drawn with respect to the validity of these methods applied to real-life situations. Second, none of the onboard electronics that would be required to actuate and control the robot were simulated. These limitations are to be addressed in future works. An additional limitation worth noting is that the link parameters and materials used in the presented simulation were not optimized in order to maximize the obtained reorientation or the feasibility of the maneuvers for the chosen motors. However, the relationship between link parameters and the resulting reorientation for a given articular trajectory can be obtained using one of the presented theoretical frameworks, with which such an optimization can be done.

A notable example of an optimization worth considering would be to consider the impact of the end pieces’ mass and length on the reorientation amplitude and motor torque requirements. Indeed, increasing the end piece inertia without changing the inertia of the remaining links will lead to higher reorientation amplitudes for a given end piece displacement, but will also increase motor torque requirements. Therefore, there is a compromise to be made, and an optimal set of parameters can most likely be found for a given robot architecture. Another important optimization that is likely to improve the performance of the presented reorientation maneuvers is to render the robot as compact as possible so as to reduce the inertia of the passive links, which, in turn, increases the induced reorientation for a given end piece displacement. However, in both cases, these optimization processes are largely dependent on the given robot architecture and the onboard hardware required to operate it and, therefore, go beyond the scope of this work.

It is also worth mentioning that although the maneuvers are well adapted to some existing legged robots, certain physical limitations, such as maximum motor acceleration and torque as well as joint limitations, may limit the amplitude of reorientation that these robots can achieve as well as the speed at which reorientation can be achieved. An obvious, yet mostly undesirable, solution is to make the required design modifications for these robots to fully implement the presented maneuvers. However, for robots of sufficient complexity, the relationship between a robot’s orientation and its articular parameters given by the presented dynamic modeling methods can be used to modify, optimize, and design maneuvers for the given context. Indeed, the maneuvers presented in this paper represent the bare-bones application of cat-inspired righting techniques to quadruped robots and, therefore, will not be optimal for every quadruped robot.

## 5 Conclusion

This paper presented a novel robot architecture with the goal to address certain shortcomings in the existing robot adaptations of the cat-righting motion, such as the limitation of reorientation to only one axis and the limited applicability of previously presented maneuvers to existing robot architectures. The proposed architecture, built as a simplified quadruped robot, features two reorientation maneuvers which allow for reorientation about the roll and pitch axes depending on the initial robot configuration. A survey of the literature on legged robots revealed that certain robots have the required appendage structure to complete the presented maneuvers. Furthermore, kinematic and dynamics simulations validated the presented maneuvers and the presented theoretical frameworks for describing free-floating robot motion. Thus, both aforementioned limitations have been addressed by the presented work. The feasibility of these maneuvers was also assessed by comparing required torque and speed values at every joint to existing motor specifications.

Further improvements to the work done in this paper will include experimental validation of the presented maneuvers as well as elaborating the proposed architecture in order to include walking maneuvers to further showcase the applicability of the presented maneuvers to existing legged robots. Experimental validation would also require work on pose detection and fall detection mechanisms in order to measure the robot’s orientation throughout the fall as well as detect when falls occur. Further work could also include the application of the presented principles to mobile robots.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable.

## Footnotes

Closed-loop movements refer to trajectories in the joint space for which the final configuration is identical to the initial configuration. Such trajectories are sometimes referred to as “cyclic trajectories.”

Following the conservation of angular momentum, the amplitude of reorientation of the robot depends on the amplitude of rotation of its end pieces and on the ratio of the inertia of the end pieces to the inertia of the rest of the robot. Since this inertia ratio is the same for both maneuvers, the difference in amplitude of reorientation between the maneuvers is obtained with different amplitudes of rotation of the end pieces.

It is worth noting that these motors are slightly larger and heavier than the *RE-max 17*. However, certain design optimizations could mitigate the effect of this, such as placing the motors that drive joints R9–R12 closer to the base of links L5–L8 and reducing the length of links L5–L8. Trajectory planning can also be optimized to reduce motor requirements, generally at the cost of smoothness of motion.

These examples do not take into account a configuration change because a $180deg$ rotation about one axis is the largest reorientation that could be required when controlling a robot’s landing configuration. Cases for which a configuration change is necessary would require a shorter sequence of maneuvers in each configuration.