Battery current measurement with shunt resistor

Question

I want to use a shunt resistor to measure the current that flows from and in a battery. The current is set by power supply for charging(positive current) and by electronic load for discharging (negative current). I want to amplify the voltage drop over the shunt resistor with instrumental op amp that gives a gain equal to 50.4. The op amp that I'm using is AD620 with Rg resistor equal to 1k with tolerance 0.1%. I want to send the output of op amp in ADC converter to acquire the measurement.

The problem is that the results aren't what I expected, shown in figure, and I don't understand which is the problem. In figure there is a comparison with hall-effect sensor(black line), instead the red line is the current measured by shunt.

^{simulate this circuit – Schematic created using CircuitLab}

The second figure shows results obtained connecting one of shunt terminal to ground

Answer 1

Would be better with Balanced sensing as shown here.

^{simulate this circuit – Schematic created using CircuitLab}