3. Simulated Kinematic Analysis

Kinematic analysis performed in MATLAB displayed the link positions, a visualization of the mechanism, and the foot trajectory. The MATLAB scripts for this analysis can be found in the appendix. This analysis uses the kinematic diagram below for vector loop equations and angle measurements from the horizontal.

To solve the mechanisms position, the MATLAB script first solves the position of links J, B, C, and K for all values of Theta M using four-bar position analysis. It then uses geometry equations to solve the positions of the other links since they are fully dependent on the position of the four-bar mechanisms in this 1 DOF mechanism. The graph below plots the angles of each link against the input angle. The links from the two four-bar sub-mechanisms are solid lines. The foot links H and I are dash-dotted lines. The rest are all dashed lines.

While this plot contains a lot of information, one trend is readily present. This mechanism is able to convert a full rotation of the input link into a motion where no links individually have to rotate that much. No link rotates more than 150 degrees from its original position. One major cause of this is the very short length of link M which makes both the four-bar sub-mechanisms meet the Grashof condition. The links 3 and 4 of the four-bar sub-mechanisms only slightly rotating is what allows the rest of the mechanism to have movement that matches the flexing of a leg more than circular motion. Using these angles as a function of the input angle, I created a simulated animation of the Jansen mechanism for one full rotation of link M.

The Grashof characteristic of the mechanism is very clear in the animation. Most links do not rotate far from their original orientation to the horizontal. This Grashof motion is converted into a walking-like motion with the geometry of the couplers and the quadrilateral. Since the couplers are triangles, their shape has 0 DOF; they cannot change shape. So coupler BDE is essentially a big kneecap, and coupler GHI is the shin which simply serves to extend downwards. The location of coupler GHI (and thus the location of the foot at point HI) is driven directly by the quadrilateral CDFG. This quadrilateral converts the motion of the two four-bars to the motion of the foot. The top four-bar AMJB defines location of link B which, through coupler BDE, defines the position of link D. The second four-bar AMKC defines the position of link C. The known positions of link D and C allow for one interior angle of the quadrilateral to be found so, knowing the length of each side, its total shape can be determined. This angle of link C relative to link D is really the key that this mechanism uses to convert the Grashof motion of two dependent four-bar mechanisms into a specific foot path. 

The MATLAB script also plotted that foot path which will be discussed in the next section.