A–H Sprawling and parasagittal position of limbs. On the left side: anterior view; on the right side: lateral view. A, B Ray-fins, attached to the middle of the body have no lever arm in relation to the body midline and therefore generate no moment. C, D Lobe-fins attached to the ventral half of the body. Note the lever arm h multiplied with the force exerted by the lobe-fin, it leads to the moment h*F which rotates the animal about its long body axis. E, F For terrestrial sprawling locomotion, two additional joints (elbow/knee, wrist/ankle) are advantageous. If the shoulder and pelvic joints are near the ventral margin of the trunk, the latter is lifted higher off the ground than if the joints were more dorsally. The autopodia can be directed medially (as in frog-like anurans) or laterally (as often in crocodiles and lizards). The elbows are directed posteriorly and the knee anteriorly. For both, the lever arms (h) of the ground reaction forces are relatively long. This leads to great torques about the proximal joints, which must be counterbalanced by contractions of the m. pectoralis in the fore- and the m. caudofemoralis in the hindlimb (Gatesy, 1999). G, H In parasagittal digitigrade limbs two further additional limb segments have been added by a mobile scapula and elongated metacarpals and -tarsals. On the right side, the limb segments are above each other, so that no or only small moments occur in anterior view. On the left side, a common position is illustrated: a joint in the middle of the freely moving limb is approaching the midline, like in the famous valgus-position of the human knee, and the typical, though less observed carpal joints of cattle, other bovids, cervids. In the hindlimb, the hock joints are in many forms approached, while the fetlocks are directed laterally. In side view (H), the upper arm as well as the lower leg pass in each step from vertical (lever arm being zero) to horizontal (lever arm reaching cosine of joint angle, that is the length of the segment. Note in E, F the centre of mass is above the supporting limbs, while in G, H it is at the level and between the scapulae/iliac blades. I, J As long as segment lengths and angles between segments are not changed, stride length does not change with either a sprawling or parasagittal posture of the limbs, and the joint moments are not bigger in sprawling limbs. sin α, excursion angle of the forelimb; sin β, excursion angle of the hindlimb; sin γ, excursion angle of the forelimb; sin δ, excursion angle of the hindlimb. Progress made is proportional to the sine of the angles. I The excursion range of the fore and hindlimb depend on rotation of the humerus and femur. J Flexion and extension of elbow and knee joint permit greater excursion angles. Fa, inertial force against being accelerated; Fi, inertial force against being retarded/braking; Fw, weight force. K In many small mammals, the upper arms and lower legs are nearly held vertically during early phases of the limb cycle, while during later phases they are swung into a nearly horizontal position (Witte et al. 1999). The lever arms (h) of the vertical ground reaction forces (GRFs) are parallel to the horizontal and their length follows the cosine of the angle between the horizontal and the upper arm/leg or the lower arm/leg. The phases in which the lever arms have their shortest or greatest lengths, and in which the moments are the smallest and biggest are shown. These moments are opposed by muscle activity. L–O Wave-like trunk movements. L An early tetrapod seen from dorsally while walking. The stylopodia are swung laterally and forward during sprawling locomotion. During the stance phase, the stylopodia compel the trunk to give way laterally and the body moves approximately in a standing wave. l, length of the stance phase. M The schematic cross section at the level of the shoulder and hip joint show the lateral displacement of the trunk. N Extension of the limb joints lift the body up and down again in a wave-like curve because parasagittal locomotion can be described by inverse pendulum mechanics (Preuschoft 2022). O GRFs acting on a lizard foot. Fx, horizontal in the direction of walking; Fy, vertical; Fz, horizontal in transverse direction (changed and redrawn after Christian, 1995). P, Q Cursorial mammals exemplified by a medium-sized antilope. l, length of the stance phase. P Fore limb (left, white) and hindlimb (right, grey) in stance phase. The lower end of the inverted pendulum is the joint closest to the ground, the upper end is the hip joint, or the pivot about which the scapula rotates. Q A fore- (left, white) and a hindlimb (right, grey) in the swing phase. Pendulum length is defined by the distance between pivot and the respective leg’s centre of mass. R Two people carrying a ladder. If both establish a firm, not mobile connection between their pelves and a ladder rung, they have to have an equal stride length and step frequency, i.e., walk in lock-step, to avoid disturbing each other. A free choice of stride length is only possible, if one of the two people loosens the connection between pelvis and rung. The plane in which the mass is moved forward is not essential, but the distance covered in the direction of locomotion.

 
 
  Part of: Preuschoft H, Krahl A, Werneburg I (2022) From sprawling to parasagittal locomotion in Therapsida: A preliminary study of historically collected museum specimens. Vertebrate Zoology 72: 907-936. https://doi.org/10.3897/vz.72.e85989