Hexapod Inverse Kinematics Equations

Hexapod Leg Layout and Leg Parts

Controlling hexapod robots requires the use of a type of mathematics called inverse kinematics. This post contains all of the inverse kinematics calculations for a hexapod robot consisting of a hexagonal body with six legs spaced equally around the perimeter. The legs are numbered one through six.

The equations in this post apply to a hexapod robot with legs spaces equally around a hexagonal frame.

Each leg on the hexapod has three servos and three “bones.” We will identify each part of the legs via analogy to the anthropod leg.

Each leg on the hexapod has three parts: coxa, femur, and tibia.

The inverse kinematics process for calculating the 18 servo angles is an actual process, with a start point, end point, and a step-by-step process to get there.

Direct Inputs

The first step is to input information about the robot geometry and commands for the robot’s movement. The geometric information is constant. This is information over which we, as the designers and pilots of the robot, have direct control.

Robot Geometry

BodySideLength == direct input
CoxaLength == direct input
FemurLength == direct input
TibiaLength == direct input

Command Inputs

PosX == control input
PosY == control input
PosZ == control input
RotX == control input
RotY == control input
RotZ == control input

Primary Geometric Calculations

From the geometry of the robot, the next step is to calculate other basic geometric relationships. Note that because this information is based on the robot’s geometry, it is constant.

Coxa Center Offsets

When we are sending commands to the hexapod to move in a particular way, those move commands refer to the position of the center of the hexapod. If we command the hexapod to move forward by 10cm, we are telling the center of the hexapod to move forward by that distance, and we compute the positions of all 18 servos on all six legs accordingly. Therefore, the first geometric calculation we will perform is to figure out the distance between the center of the hexapod, which is the origin point for our calculations, and the first (coxa) servo on each leg.

BodyCenterOffset1 = BodySideLength/2
BodyCenterOffset2 = sqrt(BodySideLength^2 - BodyCenterOffset1^2)

Body Center Offset X

Body Center Offset Y

Initial Feet Positions

All feet positions expressed with respect to the coxa servo of the corresponding leg. In other words, we basically use two different reference frames for our IK calculations. In the previous section we calculated the positions of the coxa servos of all six legs using the center of the hexapod as the origin point. Now, in this section, we will calculate the feet positions using the coxa servo of each leg as the origin point.

Leg 1

Leg 2

Leg 3

Leg 4

Leg 5

Leg 6

Body IK

The equations for each leg are of the same general form.

Leg 1

Leg 2

Leg 3

Leg 4

Leg 5

Leg 6

Leg IK

Leg 1

Leg 2

Leg 3

Leg 4

Leg 5

Leg 6

Servo Angles

Leg 1

Leg 2

Leg 3

Leg 4

Leg 5

Leg 6