Mechanical models for insect locomotion: active muscles and energy losses

 We extend the analysis of simple, energy-conserving models for the dynamics of insect locomotion in the horizontal plane developed in Schmitt and Holmes (2000a,b, 2001), where gaits characteristic of steady cockroach running and turning were evoked. In this paper, we include dissipation and energy inputs via active “muscles” in three forms: via prescribed torques at the “hip” pivot, via an active spring element of variable length, and via a pair of Hill-type muscle models representing an extensor/flexor system. Due to mechanical feedback of passive elastic forces, the stable gaits of the conservative models are preserved, and now energy input and absorption balances to additionally stabilize a preferred speed, with only modest neural sensing and feedback being required. However, these bipedal models still cannot simultaneously match observed moment-yaw magnitudes and fore-aft dynamics. Received: 17 September 2001 / Accepted: 20 February 2003 / Published online: 20 May 2003 Correspondence to: P. Holmes (e-mail: pholmes@math.Princeton.EDU) Acknowledgements. This work was supported by DARPA/ONR: N00014-98-1-0747 and DoE: DE-FG02-95ER25238. John Schmitt was partially supported by a DoD Graduate Fellowship, a Wu Fellowship of the School of Engineering and Applied Science, and a George Van Ness Lothrop Honorific Fellowship of the Graduate School at Princeton University. We thank Kenneth Meijer for allowing us to use his muscle model in Sect. 4 and Bob Full and Dan Koditschek for numerous helpful suggestions.     

Muscle activity in rapid multi-degree-of-freedom elbow movements: solutions from a musculoskeletal model

The activity of certain muscles that cross the elbow joint complex (EJC) are affected by forearm position and forearm movement during elbow flexion/extension. To investigate whether these changes are based on the musculoskeletal geometry of the joint, a three-dimensional musculotendinoskeletal computer model of the EJC was used to estimate individual muscle activity in multi-degree-of-freedom (df) rapid (ballistic) elbow movements. It is hypothesized that this model could reproduce the major features of elbow muscle activity during multi-df elbow movements using dynamic optimal control theory, given a minimum-time performance criterion. Results from the model are presented and verified with experimental kinematic and electromyographic data from movements that involved both one-df elbow flexion/extension and two-df flexion/extension with forearm pronation/supination. The model demonstrated how the activity of particular muscles is affected by both forearm position and movement, as measured in these experiments and as previously reported by others. These changes were most evident in the flexor muscles and least evident in the extensor muscles. The model also indicated that, for specific one- and two-df movements, activating a muscle that is antagonistic or noncontributory to the movement could reduce the movement time. The major features of muscle activity in multi-df elbow movements appear to be highly dependent on the joint's musculoskeletal geometry and are not strictly based on neural influences or neuroanatomical substrates. Received: 9 May 1997 / Accepted in revised form: 8 December 1998     

Analysis of a muscular control system in human movements

In this work, we have studied a muscular control system under experimental conditions for analyzing the dynamic behavior of individual muscles and theoretical considerations for elucidating its control strategy. Movement of human limbs is achieved by joint torques and each torque is specified as the sum of torques generated by muscle forces. The behavior of individual muscles is controlled by the neural input which is estimated by means of an electromyogram (EMG). In this study, the EMGs for a flexor and an extensor are measured in elbow joint movements and the dynamic behavior of individual muscles is analyzed. As a result, it is verified that both a flexor and an extensor are activated throughout the entire movement and that the activation of muscles is controlled above a specific limit independent of the hand-held load. Subsequently, a system model for simulating elbow joint movements is developed which includes the muscle dynamic relationship between the neural input and the isometric force. The minimum limit of muscle activation that has been confirmed in experiments is provided as a constraint of the neural input and the criterion is defined by a derivative of the isometric force of individual muscles. The optimal trajectories formulated under these conditions are quantitatively compared with the experimentally observed trajectories, and the control strategy of a muscular control system is studied. Finally, a muscular control system in multi-joint arm movements is discussed with regard to the comparative analysis between observed and optimal trajectories. Received: 7 April 1999 / Accepted in revised form: 27 July 1999     

Estimation of dynamic joint torques and trajectory formation from surface electromyography signals using a neural network model

In this study, human arm movement was re-constructed from electromyography (EMG) signals using a forward dynamics model acquired by an artificial neural network within a modular architecture. Dynamic joint torques at the elbow and shoulder were estimated for movements in the horizontal plane from the surface EMG signals of 10 flexor and extensor muscles. Using only the initial conditions of the arm and the EMG time course as input, the network reliably reconstructed a variety of movement trajectories. The results demonstrate that posture maintenance and multijoint movements, entailing complex via-point specification and co-contraction of muscles, can be accurately computed from multiple surface EMG signals. In addition to the model's empirical uses, such as calculation of arm stiffness during motion, it allows evaluation of hypothesized computational mechanisms of the central nervous system such as virtual trajectory control and optimal trajectory planning.     

The Influence of Intrinsic Muscle Properties on Musculoskeletal System Stability: A Modelling Study

The objective of this study is to investigate how the intrinsic mechanical properties of muscles will affect the musculoskeletal system stability. A typical musculoskeletal joint driven by a pair of antagonist muscles confined only in the sigittal plane was constructed. The dynamic characteristics of the flexor and extensor muscles induced by neural inputs were represented by three dynamic processes: neural excitation, muscle activation and muscle contraction dynamics. The muscle contraction mechanics was described using a modified Hill's model with a Contractile Element (CE), a parallel elastic element and a serial elastic element. Additionally, the change of muscle Physiological Cross-Sectional Area (PCSA) and pennation angle during muscle contraction were also considered. A set of dynamic simulations were conducted by applying an external impulsive force at the distal part of the musculoskeletal system. Sensitivity analysis was conducted to investigate the effect of the CE's force-length relationship, the CE's force-velocity relationship, the force-length relationship of the serial elastic element, the parallel elastic element and the pennation angle on the system stability. The results show that the muscles with full intrinsic mechanical properties are sufficient to stabilize the system subject to an impulsive force perturbation without reflexive changes in activations. To guarantee a self-stabilizing ability, a proper CE's force-velocity relationship, the existence of a series elastic element and a sufficient muscle co-contraction level are necessary. This study would provide insight into the intrinsic design and function of the musculoskeletal system, and also give implications for the design of bionic actuators, biomimetic robotics and prosthetic devices.     

Mathematical analysis of oxygen concentration in a two dimensional array of capillaries

 An approach is presented for modeling transport and exchange in skeletal muscle that can be used to analyze vascular beds consisting of a large number of interacting capillaries. First the oxygen concentration is determined in a functional unit consisting of a single capillary surrounded by a region of tissue in which a flux is prescribed on the outer boundary of the region. This flux, which is a result of the interaction among all of the capillaries comprising the vascular bed, is then found by matching the concentration along the borders between adjacent units. This leads to a system of ordinary differential equations for the oxygen concentration in the capillaries coupled with a system of algebraic equations for the fluxes. The method is illustrated by obtaining the oxygen concentration within an array of capillaries for the case when each capillary has a different initial concentration and for the case when each capillary has a different flow rate. Received: 12 June 2001 / Revised version: 18 April 2002 / Published online: 17 January 2003 Key words or phrases: Skeletal muscle – Transport – Microcirculation     

Trajectory formation based on physiological characteristics of skeletal muscles

Human arm trajectories in natural unrestricted reaching movements were studied. They have particular properties such that a hand path is a rather simple straight or curved line, and a tangential velocity profile of hand is bell-shaped. Also these properties are invariant, independent of movement duration and hand-held load. In this study, trajectory formation is investigated on the basis of physiological characteristics of skeletal muscles, and a criterion prescribed by a derivative of isometric muscle torque is proposed. Subsequently, optimal trajectories are formulated under various conditions of movement to account for a planning strategy of human arm trajectories. In addition to such a theoretical approach, human arm trajectories are experimentally observed by a measuring system which provides a visual sensor and a target tracking device, enabling totally unrestricted movements. Then, optimal trajectories are quantitatively evaluated in comparison with experimental data in which essential properties of human arm trajectories are demonstrated. These results support the idea that human arm trajectories are planned in order to minimize the proposed criterion which is determined from physiological aspects. Finally, the physiological advantages of human arm trajectories are discussed with regard to the analysis of observed and optimal trajectories. Received: 2 December 1997 / Accepted in revised form: 20 March 1998     

Effects of adequate vestibular stimulation on locomotor activity in fore- and hindlimb muscles in the guinea pig. Movement along a vertical axis

The effects of applying adequate vestibular stimulation to the mesencephalic locomotor region on locomotor activity in fore- and hindlimb muscles was investigated during experiments on decerebrate guinea pigs. This stimulation was produced by linear sinusoidal shifting of the animal along a vertical axis at rates of 0.08, 0.2, 0.4, and 0.8 Hz (with peak accelerations of 0.010, 0.063, 0.252, and 1.010 m·sec^–2 respectively). A downwards shift was found to increase electromyographic extensor muscle activity in fore- and hindlimbs occurring during the swing phase of the locomotor cycle. An upwards movement was accompanied by the opposite changes in muscle activity. Minimum acceleration required to produce an alteration in muscle activity equaled 0.063 m·sec^–2 (0.006g). These alterations were characterized by cyclical delay in relation to linear (active) acceleration. Phase lags in the activity of fore- and hindlimb extensor muscles at the rate of 0.8 Hz reached 63° and 86° respectively. Changes in flexor muscle activity ran counterphasically to these; phasic delay equalled 264° and 275° respectively. The part played by the vestibular system in control over locomotor activity in vertebrate muscles is discussed.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 21, No. 2, pp. 192–197, March–April, 1989.     

Effects of training and weight support on muscle activation in Parkinson’s disease

The aim of this study was to investigate the effect of high-intensity locomotor training on knee extensor and flexor muscle activation and adaptability to increased body-weight (BW) support during walking in patients with Parkinson’s disease (PD). Thirteen male patients with idiopathic PD and eight healthy participants were included. The PD patients completed an 8-week training program on a lower-body, positive-pressure treadmill. Knee extensor and flexor muscles activation during steady treadmill walking (3 km/h) were measured before, at the mid-point, and after training. Increasing BW support decreased knee extensor muscle activation (normalization) and increased knee flexor muscle activation (abnormal) in PD patients when compared to healthy participants. Training improved flexor peak muscle activation adaptability to increased (BW) support during walking in PD patients. During walking without BW support shorter knee extensor muscle off-activation time and increased relative peak muscle activation was observed in PD patients and did not improve with 8 weeks of training. In conclusion, patients with PD walked with excessive activation of the knee extensor and flexor muscles when compared to healthy participants. Specialized locomotor training may facilitate adaptive processes related to motor control of walking in PD patients.     

Power and work produced in different leg muscle groups when rising from a chair

The aim of this study was to determine the power output and work done by different muscle groups at the hip and knee joints during a rising movement, to be able to tell the degree of activation of the muscle groups and the relationship between concentric and eccentric work. Nine healthy male subjects rose from a chair with the seat at knee level. The moments of force about the hip and knee joints were calculated semidynamically. The power output (P) and work in the different muscle groups surrounding the joints was calculated as moment of force times joint angular velocity. Work was calculated as: work = f Pdt. The mean peak concentric power output was for the hip extensors 49.9 W, hip flexors 7.9 W and knee extensor 89.5 W. This power output corresponded to a net concentric work of 20.7 J, 1.0 J and 55.6 J, respectively. There was no concentric power output from the knee flexor muscles. Energy absorption through eccentric muscle action was produced by the hip extensors and hip flexors with a mean peak power output of 4.8 W and 7.4 W, respectively. It was concluded that during rising, the hip and knee muscles mainly worked concentrically and that the greatest power output and work were produced during concentric contraction of the knee and hip extensor muscles. There was however also a demand for eccentric work by the hip extensors as well as both concentric and eccentric work by the hip flexors. The knee flexor muscles were unloaded.     

Fatigue effect on cross-talk in mechanomyography signals of extensor and flexor forearm muscles during maximal voluntary isometric contractions

Objective:This paper presents the analyses of the fatigue effect on the cross-talk in mechanomyography (MMG) signals of extensor and flexor forearm muscles during pre- and post-fatigue maximum voluntary isometric contraction (MVIC).Methods:Twenty male participants performed repetitive submaximal (60% MVIC) grip muscle contractions to induce muscle fatigue and the results were analyzed during the pre- and post-fatigue MVIC. MMG signals were recorded on the extensor digitorum (ED), extensor carpi radialis longus (ECRL), flexor digitorum superficialis (FDS) and flexor carpi radialis (FCR) muscles. The cross-correlation coefficient was used to quantify the cross-talk values in forearm muscle pairs (MP1, MP2, MP3, MP4, MP5 and MP6). In addition, the MMG RMS and MMG MPF were calculated to determine force production and muscle fatigue level, respectively.Results:The fatigue effect significantly increased the cross-talk values in forearm muscle pairs except for MP2 and MP6. While the MMG RMS and MMG MPF significantly decreased (p<0.05) based on the examination of the mean differences from pre- and post-fatigue MVIC.Conclusion:The presented results can be used as a reference for further investigation of cross-talk on the fatigue assessment of extensor and flexor muscles’ mechanic.     

Motor patterns during kicking movements in the locust

Locusts (Schistocerca gregaria) use a distinctive motor pattern to extend the tibia of a hind leg rapidly in a kick. The necessary force is generated by an almost isometric contraction of the extensor tibiae muscle restrained by the co-contraction of the flexor tibiae (co-contraction phase) and aided by the mechanics of the femoro-tibial joint. The stored energy is delivered suddenly when the flexor muscle is inhibited. This paper analyses the activity of motor neurons to the major hind leg muscles during kicking, and relates it to tibial movements and the resultant forces.During the co-contraction phase flexor tibiae motor neurons are driven by apparently common sources of synaptic inputs to depolarized plateaus at which they spike. The two excitatory extensor motor neurons are also depolarized by similar patterns of synaptic inputs, but with the slow producing more spikes at higher frequencies than the fast. Trochanteral depressors spike at high frequency, the single levator tarsi at low frequency, and common inhibitors 2 and 3 spike sporadically. Trochanteral levators, depressor tarsi, and a retractor unguis motor neuron are hyperpolarized.Before the tibia extends all flexor motor neurons are hyperpolarized simultaneously, two common inhibitors, and the levator trochanter and depressor tarsi motor neurons are depolarized. Later, but still before the tibial movement starts, the extensor tibiae and levator tarsi motor neurons are hyperpolarized. After the movement has started, the extensor motor neurons are hyperpolarized further and the depressor trochanteris motor neurons are also hyperpolarized, indicating a contribution of both central and sensory feedback pathways.Variations in the duration of the co-contraction of almost twenty-fold, and in the number of spikes in the fast extensor tibiae motor neuron from 2–50 produce a spectrum of tibial extensions ranging from slow and weak, to rapid and powerful. Flexibility in the networks producing the motor pattern therefore results in a range of movements suited to the fluctuating requirements of the animal.     

Jumping mechanisms and performance in beetles. II. Weevils (Coleoptera: Curculionidae: Rhamphini)

We describe the kinematics and performance of the natural jump in the weevil Orchestes fagi (Fabricius, 1801) (Coleoptera: Curculionidae) and its jumping apparatus with underlying anatomy and functional morphology. In weevils, jumping is performed by the hind legs and involves the extension of the hind tibia. The principal structural elements of the jumping apparatus are (1) the femoro-tibial joint, (2) the metafemoral extensor tendon, (3) the extensor ligament, (4) the flexor ligament, (5) the tibial flexor sclerite and (6) the extensor and flexor muscles. The kinematic parameters of the jump (from minimum to maximum) are 530–1965 m s^?2 (acceleration), 0.7–2.0 m s^?1 (velocity), 1.5–3.0 ms (time to take-off), 0.3–4.4 μJ (kinetic energy) and 54–200 (g-force). The specific joint power as calculated for the femoro-tibial joint during the jumping movement is 0.97 W g^?1. The full extension of the hind tibia during the jump was reached within up to 1.8–2.5 ms. The kinematic parameters, the specific joint power and the time for the full extension of the hind tibia suggest that the jump is performed via a catapult mechanism with an input of elastic strain energy. A resilin-bearing elastic extensor ligament that connects the extensor tendon and the tibial base is considered to be the structure that accumulates the elastic strain energy for the jump. According to our functional model, the extensor ligament is loaded by the contraction of the extensor muscle, while the co-contraction of the antagonistic extensor and flexor muscles prevents the early extension of the tibia. This is attributable to the leverage factors of the femoro-tibial joint providing a mechanical advantage for the flexor muscles over the extensor muscles in the fully flexed position. The release of the accumulated energy is performed by the rapid relaxation of the flexor muscles resulting in the fast extension of the hind tibia propelling the body into air.     

Local innervation patterns of the metathoracic flexor and extensor tibiae motor neurons in the cricket Gryllus bimaculatus

To elucidate neural mechanisms underlying walking and jumping in insects, motor neurons supplying femoral muscles have been identified mainly in locusts and katydids, but not in crickets. In this study, the motor innervation patterns of the metathoracic flexor and extensor tibiae muscles in the cricket, Gryllus bimaculatus were investigated by differential back-fills and nerve recordings. Whereas the extensor tibiae muscle has an innervation pattern similar to that of other orthopterans, the flexor has an innervation unique to this species. The main body of the flexor muscle is divided into the proximal, middle and distal regions, which receive morphologically unique terminations from almost non-overlapping sets of motor neurons. The proximal region is innervated by about 12 moderate-sized excitatory motor neurons and two inhibitory neurons while the middle and distal regions are innervated by three and four large excitatory motor neurons, respectively. The most-distally located accessory flexor muscle, inserting on a common flexor apodeme with the main muscle, is innervated by at least four small excitatory (slow-type) and two common inhibitory motor neurons. The two excitatory and two inhibitory motor neurons that innervate the accessory flexor muscle also innervate the proximal bundles of the main flexor muscle. This suggests that the most proximal and distal parts of the flexor muscle participate synergistically in fine motor control while the rest participates in powerful drive of tibial flexion movement.     

Effects of movement speed and joint position on knee flexor torque in healthy and post-surgical subjects

Coactivation of knee flexors during knee extension assists in joint stability by exerting an opposing torque to the anterior tibial displacement induced by the quadriceps. This opposing torque is believed to be generated by eccentric muscle actions that stiffen the knee, thereby attenuating strain to joint ligaments, particularly the anterior cruciate ligament (ACL). However, as the lengths of knee muscles vary with changes in joint position, the magnitude of flexor/extensor muscle force coupling may likewise vary, possibly affecting the capacity for active knee stabilization. The purpose of this study was to assess the effect of changes in movement speed and joint position on eccentric/concentric muscle action relationships in the knees of uninjured (UNI) and post-ACL-surgery (INJ) subjects (n = 14). All subjects were tested for maximum eccentric and concentric torque of the contralateral knee flexors and extensor muscles at four isokinetic speeds (15 degrees-60 degrees x s(-1)) and four joint position intervals (20 degrees-60 degrees of knee flexion). Eccentric flexor torque was normalized to the percentage of concentric flexor torque generated at each joint position interval for each speed tested (flexor E-C ratio). In order to estimate the capacity of the knee flexors to resist active knee extension, the eccentric-flexor/concentric-extensor ratios were also computed for each joint position interval and speed (flexor/extensor E-C ratio). The results revealed that eccentric torque surpassed concentric torque by 3%-144% across movement speeds and joint position intervals. The magnitude of the flexor E-C ratio and flexor/extensor E-C increased significantly with speed in both groups of subjects (P < 0.05) and tended to rise with muscle length as the knee was extended; peak values were generated at the most extended joint position (20 degrees-30 degrees). Although torque development patterns were symmetrical between the contralateral limbs in both groups, between-group comparisons revealed significantly higher flexor/extensor E-C ratios for the INJ group compared to the UNI group (P < 0.05), particularly at the fastest speed tested (60 degrees x s(-1)). The results indicate that joint position and movement speed influence the eccentric/concentric relationships of knee flexors and extensors. The INJ subjects appeared to accommodate to surgery by developing the eccentric function of their ACL and normal knee flexors, particularly at higher speeds and at more extended knee joint positions. This may assist in the dynamic stabilization of the knee at positions where ACL grafts have been reported to be most vulnerable to strain.     

Rat hindlimb muscle blood flow during level and downhill locomotion

Duringeccentrically biased exercise (e.g., downhill locomotion), whole bodyoxygen consumption and blood lactate concentrations are lower thanduring level locomotion. These general systemic measurements indicatethat muscle metabolism is lower during downhill exercise. This studywas designed to test the hypothesis that hindlimb muscle blood flow iscorrespondingly lower during downhill vs. level exercise. Muscle bloodflow (determined by using radioactive microspheres) was measured inrats after 15 min of treadmill exercise at 15 m/min on the level (L,0°) or downhill (D, 17°). Blood flow to ankle extensormuscles was either lower (e.g., white gastrocnemius muscle: D, 9 ± 2; L, 15 ± 1 ml · min¹ · 100 g¹) or not different(e.g., soleus muscle: D, 250 ± 35; L, 230 ± 21 ml · min¹ · 100 g¹) in downhill vs. levelexercise. In contrast, blood flow to ankle flexor muscles was higher(e.g., extensor digitorum longus muscle: D, 53 ± 5; L, 31 ± 6 ml · min¹ · 100 g¹) during downhill vs.level exercise. When individual extensor and flexor muscle flows weresummed, total flow to the leg was lower during downhill exercise (D,3.24 ± 0.08; L, 3.47 ± 0.05 ml/min). These data indicate thatmuscle blood flow and metabolism are lower during eccentrically biasedexercise but are not uniformly reduced in all active muscles; i.e.,flows are equivalent in several ankle extensor muscles and higher inankle flexor muscles.     

On the reflex coactivation of ankle flexor and extensor muscles induced by a sudden drop of support surface during walking in humans.

Recent studies have revealed that the stretch reflex responses of both ankle flexor and extensor muscles are coaugmented in the early stance phase of human walking, suggesting that these coaugmented reflex responses contribute to secure foot stabilization around the heel strike. To test whether the reflex responses mediated by the stretch reflex pathway are actually induced in both the ankle flexor and extensor muscles when the supportive surface is suddenly destabilized, we investigated the electromyographic (EMG) responses induced after a sudden drop of the supportive surface at the early stance phase of human walking. While subjects walked on a walkway, the specially designed movable supportive surface was unexpectedly dropped 10 mm during the early stance phase. The results showed that short-latency reflex EMG responses after the impact of the drop (<50 ms) were consistently observed in both the ankle flexor and extensor muscles in the perturbed leg. Of particular interest was that a distinct response appeared in the tibialis anterior muscle, although this muscle showed little background EMG activity during the stance phase. These results indicated that the reflex activities in the ankle muscles certainly acted when the supportive surface was unexpectedly destabilized just after the heel strike during walking. These reflex responses were most probably mediated by the facilitated stretch reflex pathways of the ankle muscles at the early stance phase and were suggested to be relevant to secure stabilization around the ankle joint during human walking.     

Analysis of an optimal control model of multi-joint arm movements

 In this paper, we propose a model of biological motor control for generation of goal-directed multi-joint arm movements, and study the formation of muscle control inputs and invariant kinematic features of movements. The model has a hierarchical structure that can determine the control inputs for a set of redundant muscles without any inverse computation. Calculation of motor commands is divided into two stages, each of which performs a transformation of motor commands from one coordinate system to another. At the first level, a central controller in the brain accepts instructions from higher centers, which represent the motor goal in the Cartesian space. The controller computes joint equilibrium trajectories and excitation signals according to a minimum effort criterion. At the second level, a neural network in the spinal cord translates the excitation signals and equilibrium trajectories into control commands to three pairs of antagonist muscles which are redundant for a two-joint arm. No inverse computation is required in the determination of individual muscle commands. The minimum effort controller can produce arm movements whose dynamic and kinematic features are similar to those of voluntary arm movements. For fast movements, the hand approaches a target position along a near-straight path with a smooth bell-shaped velocity. The equilibrium trajectories in X and Y show an ‘N’ shape, but the end-point equilibrium path zigzags around the hand path. Joint movements are not always smooth. Joint reversal is found in movements in some directions. The excitation signals have a triphasic (or biphasic) pulse pattern, which leads to stereotyped triphasic (or biphasic) bursts in muscle control inputs, and a dynamically modulated joint stiffness. There is a fixed sequence of muscle activation from proximal muscles to distal muscles. The order is preserved in all movements. For slow movements, it is shown that a constant joint stiffness is necessary to produce a smooth movement with a bell-shaped velocity. Scaled movements can be reproduced by varying the constraints on the maximal level of excitation signals according to the speed of movement. When the inertial parameters of the arm are altered, movement trajectories can be kept invariant by adjusting the pulse height values, showing the ability to adapt to load changes. These results agree with a wide range of experimental observations on human voluntary movements. Received: 4 December 1995 / Accepted in revised form: 17 September 1996     

Estimation of instantaneous moment arms of lower-leg muscles

Muscle moment arms at the human knee and ankle were estimated from muscle length changes measured as a function of joint flexion angle in cadaver specimens. Nearly all lower-leg muscles were studied: extensor digitorum longus, extensor hallucis longus, flexor digitorum longus, flexor hallucis longus, gastrocnemius lateralis, gastrocnemius medialis, peroneus brevis, peroneus longus, peroneus tertius, plantaris, soleus, tibialis anterior, and tibialis posterior. Noise in measured muscle length was filtered by means of quintic splines. Moment arms of the mm. gastrocnemii appear to be much more dependent on joint flexion angles than was generally assumed by other investigators. Some consequences for earlier analyses are mentioned.     

Analysis of kinematic invariances of multijoint reaching movement

 There is a no unique relationship between the trajectory of the hand, represented in cartesian or extrinsic space, and its trajectory in joint angle or intrinsic space in the general condition of joint redundancy. The goal of this work is to analyze the relation between planning the trajectory of a multijoint movement in these two coordinate systems. We show that the cartesian trajectory can be planned based on the task parameters (target coordinates, etc.) prior to and independently of angular trajectories. Angular time profiles are calculated from the cartesian trajectory to serve as a basis for muscle control commands. A unified differential equation that allows planning trajectories in cartesian and angular spaces simultaneously is proposed. Due to joint redundancy, each cartesian trajectory corresponds to a family of angular trajectories which can account for the substantial variability of the latter. A set of strategies for multijoint motor control following from this model is considered; one of them coincides with the frog wiping reflex model and resolves the kinematic inverse problem without inversion. The model trajectories exhibit certain properties observed in human multijoint reaching movements such as movement equifinality, straight end-point paths, bell-shaped tangential velocity profiles, speed-sensitive and speed-insensitive movement strategies, peculiarities of the response to double-step targets, and variations of angular trajectory without variations of the limb end-point trajectory in cartesian space. In humans, those properties are almost independent of limb configuration, target location, movement duration, and load. In the model, these properties are invariant to an affine transform of cartesian space. This implies that these properties are not a special goal of the motor control system but emerge from movement kinematics that reflect limb geometry, dynamics, and elementary principles of motor control used in planning. All the results are given analytically and, in order to compare the model with experimental results, by computer simulations. Received: 6 April 1994/Accepted in revised form: 25 April 1995