Curiosity is a 3 by 2.7 by 2.3 meters, six-wheeled robot, which weighs 899 kilograms, and is destined for Gale Crater on Mars.

On a flat hard ground, it can go up to 4 cm per second, and is able to move over objects 75 cm high.

Its mission is to see if Mars ever could have supported small life forms called microbes, and if humans could survive there someday.

The Mars Science Laboratory has six wheels, each with its own individual motor, also, the two front and two rear wheels have individual steering motors (1 each). This allows the vehicle to turn in place, a full 360 degrees. The 4-wheel steering also allows the rover to swerve and curve, making arching turns.

Curiosity uses a developed version of the “Rocker-Bogie” suspension system used on its older sisters, Pathfinder and Mars Exploration Rover missions.

As we know, for a mechanism with several degrees of freedom, the equation of motion usually described by second order Lagrange equation;

**The Suspension System of the Mars rover Curiosity**

It was first introduced by, Donald B. Bickler, in 1989 under a NASA contract and with a US patent number (4,840,394)

This suspension has six wheels with symmetric structure for both sides. Each side has three wheels, which are connected to each other with two links.

Main linkage called “rocker” has two joints. While first joint connected to front wheel, other joint assembled to another linkage called “bogie,” which is similar to train wagon suspension member.

The term "rocker" comes from the design of the differential, which keeps the rover body balanced, enabling it to "rock" up or down depending on the various positions of the multiple wheels, and the term “bogie” refers to the links that have a drive wheel at each end. (Bogies were commonly used as load wheels in the tracks of army tanks as idlers distributing the load over the terrain.)

The main advantage of the rocker-bogie suspension is that the load on each wheel is nearly identical. On different positions, the wheels’ normal forces equally distribute on the contrary to four-wheel drive soft suspensions.

As the rocker-bogie design has no springs and stub axles for each wheel, it allows the rover to climb over obstacles, such as rocks, that are up to twice the wheel's diameter (50 cm) in size while keeping all six wheels on the ground. Without the springs being included in the design, the rover is able to withstand a tilt of 45 degrees in any direction without overturning. However, the rover is programmed through its "fault protection limits" in its hazard avoidance software to avoid exceeding tilts of 30 degrees during its traverses.

Friction forces and moments are considered in Lagrange motion equation for cases only when they do not depend on reaction forces of kinematic pairs. In that case, friction forces can be included in expression of equivalent forces. In vibration analysis, the friction forces can be taken into account as force, which is proportional to input velocity, q.i and introduced as dissipative function;

Using the equation above, and Lagrange motion equation, we can describe in the general form second order of motion equation of a mechanical system with several degrees of freedom by taking into account kinetic and potential energies and dissipative function as;

The rover suspension mechanism LBS consist of two symmetrical mechanisms on right and left sides of the rover body with six actuated wheels. Equation of motion, which describe the left side motion, will be valid for the right side mechanism. When rover moving on even surface, all links of the mechanism will move parallel with respect to fixed coordinates.

Assuming input rotations q7, q8 and q9 of actuators on wheels numbered 7, 8 and 9 respectively, we can write the following Lagrange equations;

Kinetic energy of mobile vehicle T can be defined by assuming that links 1, 2, 3 and 4 are in balance and mass effects at link 5 and 6 are neglected,

After a very long derivation process which I clearly do not understand fully! The final shape of the equations of motion for this three degrees of freedom system (in my defense this was not discussed in class) are given by;

This application shows that even the simplest ideas could make a huge difference in the advancement of mankind!

Thank you for your patience,

Ali Ahmad Makahleh