Stiffness Ellipse Research Articles

Virtual Stiffness: A Novel Biomechanical Approach to Estimate Limb Stiffness of a Multi-Muscle and Multi-Joint System

In recent years, different groups have developed algorithms to control the stiffness of a robotic device through the electromyographic activity collected from a human operator. However, the approaches proposed so far require an initial calibration, have a complex subject-specific muscle model, or consider the activity of only a few pairs of antagonist muscles. This study described and tested an approach based on a biomechanical model to estimate the limb stiffness of a multi-joint, multi-muscle system from muscle activations. The "virtual stiffness" method approximates the generated stiffness as the stiffness due to the component of the muscle-activation vector that does not generate any endpoint force. Such a component is calculated by projecting the vector of muscle activations, estimated from the electromyographic signals, onto the null space of the linear mapping of muscle activations onto the endpoint force. The proposed method was tested by using an upper-limb model made of two joints and six Hill-type muscles and data collected during an isometric force-generation task performed with the upper limb. The null-space projection of the muscle-activation vector approximated the major axis of the stiffness ellipse or ellipsoid. The model provides a good approximation of the voluntary stiffening performed by participants that could be directly implemented in wearable myoelectric controlled devices that estimate, in real-time, the endpoint forces, or endpoint movement, from the mapping between muscle activation and force, without any additional calibrations.

Transformation from arm joint coordinates to hand external coordinates explains non-uniform precision of hand position sense in horizontal workspace

People perceive hand position in horizontal workspace more precisely in radial than in azimuth directions and closer to the body than farther away. Current explanations for this position sense non-uniformity include spatial asymmetry in arm proprioceptive activities and/or cortex maps, experience-dependent learning, arm posture, and others. Here we investigated contributions to this non-uniformity of a posture-dependent transformation from arm joint angles, sensed by arm proprioceptors, to hand position. We measured precision of hand position sense in a bimanual hand mirror-position matching task at four horizontal targets forming a square in front of the body in 11 blindfolded individuals. We found lower hand precision in azimuth than in radial direction, higher azimuth precision at close targets, and higher radial precision at distant targets. We then theoretically analyzed the transformation of random angle errors at shoulder and elbow into hand position random errors in a horizontal plane and obtained similar distributions of hand position errors. The predicted and experimental hand-precision ellipse orientations, but not ellipse shapes or sizes, were highly correlated and were nearly orthogonal to arm stiffness ellipse orientations reported in the literature. We concluded that the joint-to-hand coordinate transformation is responsible for the non-uniform precision of hand position sense.

Joint stiffness of a musculoskeletal bionic leg based on the foot stiffness ellipse model

For the bionic jumping robot, there is contact force between the foot and the environment inevitably. Bionic jumping robots need to interact with the environment for force/position information in a compliant manner. Therefore, it is necessary to analyze the joint stiffness and foot stiffness of a bionic jumping robot. Based on analyzing the muscle arrangement and flexible jumping movement principle of a dog leg, a kind of musculoskeletal leg driven by pneumatic artificial muscles (PAMs) is presented. The centroid trajectory of the bionic leg is planned for jumping. Each joint stiffness is derived by the joint torque, which is changing with PAM inner pressure and joint angular displacement. On the other hand, joint stiffness can be planned by the foot stiffness. A kind of ellipse stiffness model of foot is proposed by analyzing the foot elastic potential energy caused by the contact force, and each joint stiffness of the bionic leg is calculated by the foot stiffness, which is mapping by the Jacobian matrix. The joint angular stiffness is analyzed by changing the foot stiffness ellipse parameters, and the expected joint stiffness of the bionic leg for jumping is planned based on the foot stiffness model. This study will pay a foundation for controlling the joint stiffness to achieve stable jumping of the bionic leg.

The Concurrent Control of Motion and Contact Force in the Presence of Predictable Disturbances.

The simultaneous control of force and motion is important in everyday activities when humans interact with objects. While many studies have analyzed the control of movement within a perturbing force field, few have investigated its dual aspects of controlling a contact force in nonisometric conditions. The mechanism by which the central nervous system controls forces during movements is still unclear, and it can be elucidated by estimating the mechanical properties of the arm during tasks with concurrent motion and contact force goals. We investigate how arm mechanics change when a force control task is accomplished during low-frequency positional perturbations of the arm. Contrary to many force regulation algorithms implemented in robotics, where contact impedance is decreased to reduce force fluctuations in response to position disturbances, we observed a steady increase of arm endpoint stiffness as the task progressed. Based on this evidence, we propose a theoretical framework suggesting that an internal model of the perturbing trajectory is formed. We observed that force regulation in the presence of predictable positional disturbances is implemented using a position control strategy together with the modulation of the endpoint stiffness magnitude, where the direction of the endpoint stiffness ellipse's major axis is oriented toward the desired force.

Introducing Whole Finger Effects in Surface Haptics: An Extended Stick-Slip Model Incorporating Finger Stiffness.

The kinematic serial chain configuration of a finger modulates the frictional properties during tactile exploration tasks. This paper analyzes and subsequently models the effects of the entire finger during sliding operations on a surface. Qualitative and quantitative study of finger movement patterns with postures, sliding directions, and contact angles first indicate the effect of finger stiffness on contact mechanics. A "stiffness ellipse" is subsequently modeled to incorporate finger pose effects, and then coupled with the lumped mass-spring-damper model of the finger pad to estimate resultant contact forces. The performance of the proposed model is verified by comparing with experimental results obtained from ten subjects. The proposed model could estimate the general tendencies of contact forces with change in postures (Extended and Flexed), sliding directions (proximal and distal), and contact angles (20°, 40° and 60°). The experimental results indicate that finger stiffness significantly modulates the contact forces, stick-slip frequency, preloading duration and initial spike during sliding. Introduction of finger posture effects could explain the change in finger normal force during tactile exploration tasks. The proposed haptic rendering model can be used to give a more natural user feedback in virtual fingertip-surface interactions.

Joint variable stiffness of musculoskeletal leg mechanism for quadruped robot

When the quadruped robot is in locomotion such as jumping and running with higher speed, there is non-continuous contact force between the foot and the environment inevitably. In order to achieve the flexible force interaction of the bionic legs with the environment, it is necessary to analyze the joint angular stiffness of the bionic leg. In this article, based on the designing principles of the bionics, light-weighted, and flexible, a kind of musculoskeletal bionic leg mechanism driven by pneumatic artificial muscles is presented by inspiring from the biological cheetah anatomy and physiology muscle distribution. The kinematics of the bionic leg is analyzed to obtain the Jacobian matrix. In order to achieve high-speed jumping and soft landing of the bionic leg, a kind of foot stiffness model is presented by analyzing the foot elastic potential energy caused by the contact force. The mapping relationship between the joint stiffness matrix and the foot stiffness matrix is obtained by the Jacobian matrix. Then, the dynamics of the bionic leg is analyzed to determine the relationship between the joint angular stiffness and the pneumatic artificial muscle gas pressure. Finally, the experiment on controlling the pneumatic artificial muscles gas pressure for tracking the joint angular stiffness of the bionic leg is conducted. By regulating the pneumatic artificial muscle gas pressure, the needed pneumatic artificial muscle gas pressure that could meet the desired foot stiffness ellipse model can be determined. The study will pay a theoretical foundation for controlling the pneumatic artificial muscles to achieve the high-speed locomotion of the bionic leg.

Pilot study on quantitative assessment of muscle imbalance: differences of muscle synergies, equilibrium-point trajectories, and endpoint stiffness in normal and pathological upper-limb movements.

This paper proposes a novel method for assessment of muscle imbalance based on muscle synergy hypothesis and equilibrium point (EP) hypothesis of motor control. We explain in detail the method for extracting muscle synergies under the concept of agonist-antagonist (AA) muscle pairs and for estimating EP trajectories and endpoint stiffness of human upper limbs in a horizontal plane using an electromyogram. The results of applying this method to the reaching movement of one normal subject and one hemiplegic subject suggest that (1) muscle synergies (the balance among coactivation of AA muscle pairs), particularly the synergies that contributes to the angular directional kinematics of EP and the limb stiffness, are quite different between the normal subject and the hemiplegic subject; (2) the concomitant EP trajectory is also different between the normal and hemiplegic subjects, corresponding to the difference of muscle synergies; and (3) the endpoint (hand) stiffness ellipse of the hemiplegic subject becomes more elongated and orientation of the major axis rotates clockwise more than that of the normal subject. The level of motor impairment would be expected to be assessed from a comparison of these differences of muscle synergies, EP trajectories, and endpoint stiffness among normal and pathological subjects using the method.

Strategy switching in the stabilization of unstable dynamics.

In order to understand mechanisms of strategy switching in the stabilization of unstable dynamics, this work investigates how human subjects learn to become skilled users of an underactuated bimanual tool in an unstable environment. The tool, which consists of a mass and two hand-held non-linear springs, is affected by a saddle-like force-field. The non-linearity of the springs allows the users to determine size and orientation of the tool stiffness ellipse, by using different patterns of bimanual coordination: minimal stiffness occurs when the two spring terminals are aligned and stiffness size grows by stretching them apart. Tool parameters were set such that minimal stiffness is insufficient to provide stable equilibrium whereas asymptotic stability can be achieved with sufficient stretching, although at the expense of greater effort. As a consequence, tool users have two possible strategies for stabilizing the mass in different regions of the workspace: 1) high stiffness feedforward strategy, aiming at asymptotic stability and 2) low stiffness positional feedback strategy aiming at bounded stability. The tool was simulated by a bimanual haptic robot with direct torque control of the motors. In a previous study we analyzed the behavior of naïve users and we found that they spontaneously clustered into two groups of approximately equal size. In this study we trained subjects to become expert users of both strategies in a discrete reaching task. Then we tested generalization capabilities and mechanism of strategy-switching by means of stabilization tasks which consist of tracking moving targets in the workspace. The uniqueness of the experimental setup is that it addresses the general problem of strategy-switching in an unstable environment, suggesting that complex behaviors cannot be explained in terms of a global optimization criterion but rather require the ability to switch between different sub-optimal mechanisms.

Musculotendon translational stiffness and muscle activity are modified by shear forces

BackgroundIn this study we investigate whether the nervous system is responsive to anterior–posterior shearing forces applied to the knee. Specifically, we examine whether the nervous system will increase musculotendon stiffness along the anterior–posterior degree-of-freedom via relative changes in muscle activation. MethodsWhile seated on a Biodex, 12 male participants performed knee extensor moments against a cuff at three different distances (265mm, 310mm, and 355mm) from the knee joint. This was done for each combination of knee angle (20°, 45°, 70°), moment (25%, 75% of maximum), and angular velocity (0°/s, 10°/s). By altering cuff position, the magnitude of the shear force could be changed while holding moment constant. We then calculated the 6-degree-of-freedom, musculotendon stiffness matrix of the knee and interpreted the eigenvalues and eigenvectors of this matrix using stiffness ellipses. FindingsAnterior–Posterior knee stiffness significantly increased with an increase in shear force at 20° and 45° of knee flexion. This coincided with significant increases in semitendinosus and vastus medialis activity. InterpretationWe found that the nervous system can significantly increase stiffness along a translational degree-of-freedom in response to translational loading, and did so by activating muscles geometrically oriented to provide stiffness along the anterior–posterior degree-of-freedom.

Hopping by a Monopedal Robot with a Biarticular Muscle by Compliance Control – An Application of an Electromagnetic Linear Actuator –

The compliance of muscles with external force and the structural stability given by biarticular muscles are important features of animals for realizing dynamic whole-body motion such as running and hopping in various environments. For this reason, we have been studying an electromagnetic linear actuator. This actuator emulates the behavior of a human muscle, such as spring-damper properties, through the quick control of output force, i.e., impedance control. It is expected to be used as an artificial muscle. In this paper, we design a monopedal robot possessing bi- and mono-articular muscles implemented by linear actuators. Thanks to the biarticular muscle, the direction of bouncing by a robot can be controlled by changing the stiffness ellipse at the endpoint, i.e., foot, of the robot. We make a simulator of the robot to evaluate dynamic characteristics and show that the robot hops stably by adjusting the stiffness ellipse. We also confirm that the behavior of the real robot is consistent with that of our simulator.

Control of a Manipulator with Bi-Articular Muscles (Simultaneous Control of Trajectory and Stiffness Ellipse Using Saturation Functions)

In a control method of a manipulator, it is important that an end effector can softly touch a subject. In order to realize a control for flexibility of a hand, a control with an artificial muscle actuator has been executed. However, the artificial muscle actuators are known to have a problem in controllability because they have nonlinear viscoelastic properties. This study shows simultaneous control of manipulator posture and joint stiffness using the artificial muscle actuators. It is realized by using a simple feedback control system. We also deal with the method to obtain the ellipse that indicates the stiffness characteristic of the end effector. Moreover, the control method is used for leg control of a walking robot. The control performance of the proposed method is verified by simulation.

Effects of Arm Stiffness and Muscle Effort on Position Reproduction Error in the Horizontal Plane

present study investigated the effects of two-dimensional arm stiffness and muscle effort required to maintain horizontal arm posture on position-reproduction errors. 12 participants performed a multi-joint position-reproduction task without visual feedback. They were required to indicate a proprioceptively remembered target position with the fingertip of the ipsilateral arm. The results showed that both constant and variable errors were larger in the direction of lower stiffness rather than in the direction of higher stiffness in the stiffness ellipse. In the condition where participants' arm was supported during position perception, variable error was larger than when it was vertically unsupported. These results suggested that proprioceptive accuracy and precision are positively related to the axis length of elliptically represented arm stiffness, and that exerting muscle effort to maintain the arm against the force of gravity may be supportive of human proprioceptive mechanisms.

The passive stiffness of the wrist and forearm

Because wrist rotation dynamics are dominated by stiffness (Charles SK, Hogan N. J Biomech 44: 614-621, 2011), understanding how humans plan and execute coordinated wrist rotations requires knowledge of the stiffness characteristics of the wrist joint. In the past, the passive stiffness of the wrist joint has been measured in 1 degree of freedom (DOF). Although these 1-DOF measurements inform us of the dynamics the neuromuscular system must overcome to rotate the wrist in pure flexion-extension (FE) or pure radial-ulnar deviation (RUD), the wrist rarely rotates in pure FE or RUD. Instead, understanding natural wrist rotations requires knowledge of wrist stiffness in combinations of FE and RUD. The purpose of this report is to present measurements of passive wrist stiffness throughout the space spanned by FE and RUD. Using a rehabilitation robot designed for the wrist and forearm, we measured the passive stiffness of the wrist joint in 10 subjects in FE, RUD, and combinations. For comparison, we measured the passive stiffness of the forearm (in pronation-supination), as well. Our measurements in pure FE and RUD agreed well with previous 1-DOF measurements. We have linearized the 2-DOF stiffness measurements and present them in the form of stiffness ellipses and as stiffness matrices useful for modeling wrist rotation dynamics. We found that passive wrist stiffness was anisotropic, with greater stiffness in RUD than in FE. We also found that passive wrist stiffness did not align with the anatomical axes of the wrist; the major and minor axes of the stiffness ellipse were rotated with respect to the FE and RUD axes by ∼20°. The direction of least stiffness was between ulnar flexion and radial extension, a direction used in many natural movements (known as the "dart-thrower's motion"), suggesting that the nervous system may take advantage of the direction of least stiffness for common wrist rotations.

Stabilization Strategies for Unstable Dynamics

BackgroundWhen humans are faced with an unstable task, two different stabilization mechanisms are possible: a high-stiffness strategy, based on the inherent elastic properties of muscles/tools/manipulated objects, or a low-stiffness strategy, based on an explicit positional feedback mechanism. Specific constraints related to the dynamics of the task and/or the neuromuscular system often force people to adopt one of these two strategies.Methodology/FindingsThis experiment was designed such that subjects could achieve stability using either strategy, with a marked difference in terms of effort and control requirements between the two strategies. The task was to balance a virtual mass in an unstable environment via two elastic linkages that connected the mass to each hand. The dynamics of the mass under the influence of the unstable force field and the forces applied through the linkages were simulated using a bimanual, planar robot. The two linkages were non-linear, with a stiffness that increased with the amount of stretch. The mass could be stabilized by stretching the linkages to achieve a stiffness that was greater than the instability coefficient of the unstable field (high-stiffness), or by balancing the mass with sequences of small force impulses (low-stiffness). The results showed that 62% of the subjects quickly adopted the high-stiffness strategy, with stiffness ellipses that were aligned along the direction of instability. The remaining subjects applied the low-stiffness strategy, with no clear preference for the orientation of the stiffness ellipse.ConclusionsThe choice of a strategy was based on the bimanual coordination of the hands: high-stiffness subjects achieved stability quickly by separating the hands to stretch the linkages, while the low-stiffness subjects kept the hands close together and took longer to achieve stability but with lower effort. We suggest that the existence of multiple solutions leads to different types of skilled behavior in unstable environments.

非線形バネを用いた腱駆動機構の剛性楕円制御

Versatile and safe manipulators are required for use in human environments. A tendon mechanism with nonlinear springs that can mechanically adjust joint stiffness is one candidate that can satisfy the requirements of versatility and safety. In this paper, a hybrid stiffness ellipse control for controlling both mechanical and control stiffness at the end of the manipulator is proposed for the tendon mechanisms. This study shows the transformation of the stiffness ellipse at the ends of arms to joint stiffness. The transformation is applied to both mechanical joint stiffness and controller gain of joint angle controller. Simulation results verify the validity of the proposed method.

1P1-T07 剛性楕円の調整による柔軟な物理インタラクションを行う筋骨格上肢ロボットの開発(バイオロボティクス)

Development of mechanisms for robots in daily life to realize safe human-robot interaction is important issue of robotics. Humans have two important mechanisms to move in various environments while coping with unexpected disturbances. One is the compliance of muscles, and the other is the biarticular muscle. These mechanisms seem to be necessary to realize human like compliant motion. In this research, electromagnetic linear actuators are used as artificial muscles, because it is easy to implement multi-articular muscles and they can change the impedance characteristics by controlling their output force. We show the effect of biarticular muscles for the controlling of the stiffness ellipse at the end effector and develop a robotic arm with 3 actuators. The robot can change the stiffness ellipse by controlling the stiffness coefficient of each actuator and perform reaching tasks.

2A2-D03 筋骨格モデルと筋電計測に基づく上肢運動の運動評価(筋骨格モデリング)

Humans control the stiffness of joints depending on type of the target task. This implies that profiles of the stiffness includes fruitful information about skillfulness of the human performing. The purpose of this research is to analyze motions by estimating the stiffness and equilibrium points of joints based on sEMG measurements and musculoskeletal model. In this paper, the proficiency evaluation method is proposed based on the calculation of the stiffness ellipse and the trajectory of the equilibrium points of joints.

‘Brute force’vs.‘gentle taps’ in the control of unstable loads

‘Brute force’<i>vs.</i>‘gentle taps’ in the control of unstable loads

2A2-E03 弾性楕円に基づくリニア電磁アクチュエータ駆動型一脚ロボットの制御(脚移動ロボット)

We have been studying an interior permanent magnet linear actuator as an artificial muscle. This actuator has long stroke, high response and large thrust using interior permanent magnets effectively and can emulate the spring-dumper characteristics of a human muscle by quick control of the output force, i.e. impedance control. In this paper, we develop a monopedal robot possessing bi- and mono-articular muscles implemented by electromagnetic linear actuators. Thanks to the biarticular muscle, the stiffness ellipse at the endpoint (foot) of the robot can be changed. The bouncing direction of the robot can be controlled by adjusting the stiffness ellipse when the robot touches down to the ground. We made a simulator of the robot to evaluate the dynamic characteristics. We also confirm that the behavior of the real robot is consistent to that of our simulator.

Toward the Design of a Decoupled, Two-Dimensional, Vision-Based μN Force Sensor

In this paper, we present three designs for a decoupled, two-dimensional, vision-based micro-Newton (μN) force sensor for microrobotic applications. There are currently no reliable, multi-axis, commercially-available force sensors to measure forces at this scale that can be easily integrated into standard microrobotic test-beds. In our previous work, we presented a design consisting of a planar, elastic mechanism with known force-deflection characteristics. It was inspired by the designs of pre-existing micro electromechanical system suspension mechanisms. A charge-coupled device camera was used to track the deformation of the mechanism as it was used to manipulate objects in a microscale/mesoscale robotic manipulation test-bed. By observing the displacements of select points on the mechanism, the manipulation forces were estimated. In this work, we have designed a compliant mechanism with decoupled stiffness using the building block approach. By designing mechanisms with circular compliance and stiffness ellipses along with zero magnitude compliance and stiffness vectors, we are able to achieve our design requirements. Validation of this approach through the testing of macroscale prototypes and a scaled design for microrobotic applications are offered, along with a sensitivity analysis, yielding insights for microfabricating such designs.